mjhopkins.github.iomjhopkins.github.io/files/thesis.pdfintroduction the foundations of the field of...

SID

ERE·MEN

S·EADEM

·MUTAT

O

University of Sydney

School of Mathematics and Statistics

Quantum affine algebras:

quantum Sylvester theorem,skew modules and centraliser construction.

Mark J. Hopkins

A dissertation submitted in partial fulfilment

of the requirements of the degree of PhD (Mathematics).

2007

Introduction

The foundations of the field of quantum groups arose in the work of the St. Petersburg

(Leningrad) school of mathematical physicists around L. D. Faddeev in the late 1970’s

and early 1980’s in relation with the quantum inverse scattering method (QISM); see e.g.

Reshetikhin, Takhtajan and Faddeev [RTF] for the basics of this approach, which involves

the study of integrable models in quantum statistical mechanics. (See also section 7.5 of

[CP] for an explanation of how the study of integrable models for 2-dimensional lattice

models leads to the quantum Yang–Baxter equation and quantum algebras such as Y(sl2)).

The methods of the quantum inverse scattering problem lead to equations of the form

RT1T2 = T2T1R

R12R13R23 = R23R13R12(0.1)

where T is an n×n matrix with elements taken from some algebra A, and R =∑rijkleij⊗

ekl ∈ Mn2(C)⊗Mn2(C). The notations Rij indicate in which copies of Mn2(C) the matrixR acts, that is R12 = R ⊗ 1, R23 = 1 ⊗ R,R13 =

∑rijkleij ⊗ 1 ⊗ ekl. The first equation

here is the RTT or ternary relation, while the second is the celebrated quantum Yang–

Baxter equation. See for instance Takhtajan and Faddeev (1979), Kulish and Sklyanin

(1982), Kulish and Reshetikhin (1986). This matrix form of the defining relations for the

Yangian Y(gln) and quantum affine algebra Uq(gln) (see below) allows special algebraic

techniques (the so-called R-matrix formalism) to be used to describe the structure and

to study representations of these algebras. This R-matrix approach also leads to a clear

description of the Hopf algebra structure of these algebras.

We develop the R-matrix techniques as a powerful instrument to investigate the struc-

ture of these algebras. The key results there are the constructions of a special formal series

called the quantum determinant .

The Yangians were originally regarded primarily as a vehicle for producing rational

solutions of the Yang–Baxter equation; cf. Drinfeld (1985). Conversely, the ternary relation

is a powerful tool for studying quantum groups themselves; see e.g. Reshetikhin, Takhtajan

and Faddeev (1990).

The fundamental paper (1985) of V. G. Drinfeld [D1] marked the beginning of the field

of quantum groups , in which these algebras are studied as objects of purely mathematical

interest. See also [D2]. In this paper he introduced the term Yangian (in honour of

C. N. Yang), and also defined the quantized Kac–Moody algebras , which were introduced

iii

iv INTRODUCTION

independently by M. Jimbo (1985). He also realised that these algebras had a natural Hopf

algebra structure.

The Yangians defined by Drinfeld form a remarkable family of quantum groups related

to rational solutions of the classical Yang–Baxter equation. Drinfeld and Jimbo’s definition

generalised the Yangian for the general (or special) linear Lie algebra, which had been

studied previously in the St. Petersburg school. For each simple finite-dimensional Lie

algebra a over the field C of complex numbers, the corresponding Yangian is defined asa canonical deformation of the universal enveloping algebra U(a [z]) for the polynomial

current Lie algebra a [z]. Importantly, the deformation is considered in the class of Hopf

algebras, which guarantees its uniqueness under some natural homogeneity conditions.

Another presentation of the Yangian for a was given later by Drinfeld (1988) [D3]. The

algebra which is now called the Yangian for the general linear Lie algebra glN and denoted

by Y(glN), had been considered a few years earlier in the work of L. D. Faddeev and the

St.-Petersburg school. The defining relations of the Yangian Y(glN) can be written in the

form of a single ternary (or RTT ) relation on the matrix of generators, allowing the use

of R-matrix techniques. However, for general a the Yangian Y(a) does not admit such a

convenient presentation.

From the algebraic point of view, the algebra Y(gln) (and the closely related Yangian

Y(sln) for the special linear Lie algebra sln which is the quotient of Y(gln) by its center) is

exceptional in the following sense. For any simple Lie algebra a , the corresponding Yangian

contains the universal enveloping algebra U(a) as a subalgebra. However, only in the case

a = sln does there exists a homomorphism from the Yangian for a to U(a) (the evaluation

homomorphism) which is identical on the subalgebra U(a) (Drinfeld, 1985). This property

plays a key role in the applications of the Yangians to conventional representation theory.

In this dissertation we concentrate on this distinguished algebra Y(gln).

The quantum affine algebras form an important family of quantized enveloping algebras.

They were introduced by Drinfeld and Jimbo in 1985 in the general context of quantum

groups and have since been studied by many authors; see e.g. the books by Chari and

Pressley [CP] or Klimyk and Schmudgen [KS] for a detailed account of their origins and

applications. Once again, the particular RTT -presentation of the A-series quantum affine

algebras was originally introduced and studied in the work of L.D. Faddeev and the St.-

Petersburg school. However this form of presentation does not exist in this convenient

form for more general quantum affine algebras.

The representation theory of the quantum affine algebras and Yangians, as well as

various applications of the theory, are discussed in the mentioned above book [CP, Chap-

ter 12]. Amongst other applications we note a surprising connection of the QISM I and

QISM II algebras with the hypergeometric functions discovered by Kuznetsov and Koorn-

winder [KK, Ku].

INTRODUCTION v

The flurry of activity in studying these algebras which followed the initial papers of

Drinfeld and Jimbo has led to applications in many areas of mathematics, for example in

classifying links in knot theory and producing invariants of 3-manifolds in low-dimensional

topology. Quantum groups also play an important role in Alain Connes’ program for a

theory of non-commutative geometry. In physics these algebras, and their super analogues,

have found applications in, amongst others, the theory of integrable models in statistical

mechanics, in conformal field theory and quantum gravity.

The Yangian can be regarded as a “degenerate” version of the quantum affine algebra

in which the deformation parameter q has been scaled away (see [D1], [To]).

Note also that an analogue of the Frobenius–Schur duality holds for the Yangian and

quantum affine algebra. The classical result is an equivalence of monoidal categories be-

tween a category of right Sl-modules and a category of left sln-modules. The quantum

affine algebra is dual in this sense to the affine Hecke algebra, while the Yangian is dual

to the degenerate affine Hecke algebra; see [CP, Section 12.3]. This helps to explain the

close parallel between the structure and representation theory of these algebras.

In this regard, we note that V. Tolstoy has defined the Drinfeldians [To, To2] which

form a class of algebras depending upon two parameters which simultaneously generalise

the Yangians and quantum affine algebras.

We now discuss the contents of this dissertation in greater detail. For any indices

i, j = 1, . . . , n introduce the minors cij of A corresponding to the rows 1, . . . ,m,m+ i and

columns 1, . . . ,m,m + j so that

cij = a1...m,m+i1...m,m+j.

Let A(m) be the submatrix of A determined by the first m rows and columns. The classical

Sylvester theorem (see for example [T] or [L]) provides a formula for the determinant of

the matrix C = [cij] :

detC = detA ∙(detA (m)

)n−1.

A general noncommutative analogue of the Sylvester theorem was given by Gelfand and

Retakh [GR], where the numerical entries of the matrix A are replaced by entries from a

non-commutative ring. This analogue relies on the theory of quasideterminants originated

by these authors; see also Gelfand and Retakh [GR2], Gelfand et al. [GKLLRT, GGRW].

Some versions of the Sylvester theorem for various quantum matrices were given by Krob

and Leclerc [KL] and Konvalinka [K]. We shall prove and make use of analogues of this

theorem in which the matrix A is replaced by the matrix of generators of the Yangian

Y(gln) and the quantum affine algebra Uq(gln).

By the results of Olshanski [O1, O2], the A type Yangian is recovered as a certain

projective limit of centralizers in the universal enveloping algebras. It is remarkable that

a quantum group such as the Yangian appears naturally in a construction involving only

classical algebras Let n and m be integers such that n > m > 0. Denote by Am(n) the

vi INTRODUCTION

centralizer of gln−m in the universal enveloping algebra U(gln), where gln−m is regarded as

a natural subalgebra of gln. For any fixed m there exists a chain of natural homomorphisms

Am(m)← Am(m+ 1)← ∙ ∙ ∙ ← Am(n)← ∙ ∙ ∙ (0.2)

which respect the natural filtrations inherited from the universal enveloping algebra. So,

one can define the corresponding projective limit algebra

Am = limprojAm(n), n→∞ (0.3)

in the category of filtered algebras. In particular, the algebra A0 is the projective limit of

the centers of U(gln) and it is isomorphic to an algebra of polynomials in countably many

variables. Moreover, for m > 0 one has the isomorphism [O1, O2]:

Am ∼= A0 ⊗ Y(glm), (0.4)

where Y(glm) is the Yangian for the Lie algebra glm.

A key part of the proof of the decomposition (0.4) is a construction of algebra homo-

morphisms

Y(glm)→ Am(n), n = m,m+ 1, . . . (0.5)

compatible with the chain (0.2) which define an embedding

Y(glm) ↪→ Am. (0.6)

A similar construction can be applied to the orthogonal and symplectic Lie algebras

which leads to the introduction of the corresponding twisted Yangians; see Olshanski [O3]

and Molev and Olshanski [MO]. Super analogues of the centralizer construction were

given by Nazarov [N] and Nazarov and Sergeev [NS] to obtain the Yangian for the queer

Lie superalgebra.

In this dissertation, we develop a q-analogue of the Olshanski construction where the

role of the Yangian is played by the q-Yangian, a subalgebra of the quantum affine algebra.

This enables us to give a new proof of the Poincare–Birkhoff–Witt theorem for Uq(gln).

Moreover, the homomorphism (0.5) gives rise to a representation of the Yangian in

the homomorphism space Homglm(L(μ), L(λ)

)which is called the skew (or elementary)

representation. Such representations were studied by Cherednik [C] (mainly in the context

of the quantum affine algebras) and by Nazarov and Tarasov [NT2]; see also Molev [Mo],

Khoroshkin and Nazarov [KN1, KN2].

Throughout this dissertation we will study the algebraic structure and representation

theory of the Yangian Y(gln) in parallel with that of the quantum affine algebra Uq(gln).

These algebras are similar in many regards, but the quantum affine algebra presents greater

complications owing to the appearance of powers of the parameter q and the presence of two

matrices of generators in the presentation we shall employ, rather than one as for Y(gln).

We will include the details for the simpler case of the Yangian as many calculations in the

q-case are quite similar and require just minor corrections.

INTRODUCTION vii

In Chapter 1 we provide definitions and basic properties of the algebras of interest. We

have two pairs of algebras, that is the pair U(gln) and Y(gln), and the pair Uq(gln) and

Uq(gln). In both cases the first is generated by a matrix T subject to relations

RT1T2 = T2T1R

and the second by the coefficients of a matrix T (u) subject to relations

R(u, v)T1(u)T2(v) = T2(v)T1(u)R(u, v).

In both cases there is an inclusion of the first algebra into the second and an evaluation

homomorphism from the second onto the first.

We describe some automorphisms of these algebras and specify in what sense we can

regard Uq(gln) as specialising to U(gln) as q → 1. We also define the quasi-determinantsof a non-commutative algebra.

We then develop aspects of the structure theory of these algebras, first for the Yangian,

then for the quantum affine algebra.

The notion of ‘quantum determinant’ was originally due to Kulish and Sklyanin [KSk].

In Chapter 2 we define the quantum determinant of the Yangian Y(gln), whose coefficients

are central in Y(gln).

In Chapter 3 we prove an analogue of the Sylvester theorem for the generator matrices

of the Yangian.

Chapter 4 we apply the quantum Sylvester theorem for Y(gln) to produce a new proof

of Olshanski’s result that the Yangian Y(glm) is contained as a subalgebra in the projective

limit of certain centralisers in U(gln) cf. [Mo], [O2]. This proof is interesting in that it

relies only on constructions in linear algebra. We would like to thank Serge Ovsienko

for help with the proof of Theorem 4.9. As a corollary to this we obtain a new proof of

the Poincare–Birkhoff–Witt theorem for Y(gln). This theorem for general Yangians was

known to Drinfeld although he did not publish a proof. For other proofs see Brundan and

Kleshchev [BK1], Nazarov [N], Olshanski [O2]. A proof of the Poincare–Birkhoff–Witt

theorem for all Drinfeld Yangians was given by Levendorskii [Le].

We then move to the q-case. In Chapter 5, we introduce q-permutations and q-

antisymmetrizers and their basic properties, and use these to define and study the quantum

determinants of the quantum affine algebra Uq(gln), whose coefficients again are central.

In Chapter 6 we prove an analogue of the Sylvester theorem for the generator matrices

of the quantum affine algebra Uq(gln).

In Chapter 7 we apply a homomorphism associated with the quantum Sylvester the-

orem to give a q-analogue of the centralizer construction where U(gln) is replaced by the

quantized enveloping algebra Uq(gln), and the Yangian Y(glm) with the q-Yangian Yq(glm),

which is a natural subalgebra of the quantum affine algebra Uq(glm). That is, we construct

a q-analogue of the Olshanski algebra Am as a projective limit of certain centralizers in

viii INTRODUCTION

Uq(gln) and show that this limit algebra contains the q-Yangian as a subalgebra (see The-

orem 7.4 below). As a corollary we obtain a new proof of the Poincare–Birkhoff–Witt

theorem for Uq(gln). We follow the approach of [Mo] where an alternative embedding

(0.6) was constructed. It is based upon a version of the quantum Sylvester theorem (see

Theorem 6.6 below); cf. Gelfand and Retakh [GR], Krob and Leclerc [KL].

We believe that a q-version of the isomorphism (0.4) also takes place, however, we do

not have a proof.

We then turn to representation theory. In Chapters 8 and 9 we use homomorphisms

associated with the quantum Sylvester theorems for the algebras Y(gln) and Uq(gln) to give

an explicit realizations of their skew representations. We generally follow the approach

of [Mo], simplifying the arguments with the use of some observations of Brundan and

Kleshchev [BK1]. The construction allows us to calculate in a particularly simple way their

highest weight, Drinfeld polynomials and Gelfand–Tsetlin characters. In the Yangian case,

the Drinfeld polynomials were calculated by Nazarov and Tarasov [NT2]; see also [BR],

[C], [NN]. In a more general context, the Gelfand–Tsetlin characters were introduced by

Brundan and Kleshchev [BK2] in their study of representations of the shifted Yangians.

They are analogous to the Yangian characters of Knight [Kn] and the q-characters of

Frenkel and Reshetikhin [FR] for the quantum affine algebras.

I would like express my gratitude for the infinite patience

and forbearance, and the unstinting support of my supervisor

Dr. Alex Molev.

Contents

Introduction iii

Chapter 1. Definitions and preliminaries 1

1.1. General definitions 1

1.2. The enveloping algebra U(gln) 2

1.3. The Yangian Y(gln) 3

1.4. The quantum enveloping algebra Uq(gln) 6

1.5. Quantum affine algebra 9

1.6. Quasi-determinants 11

Chapter 2. Quantum determinant of Y(gln) 15

Chapter 3. Quantum Sylvester theorem for Y(gln) 23

Chapter 4. Centraliser construction for U(gln) 29

Chapter 5. Quantum determinant for Uq(gln) 39

5.1. Fundamental relation 39

5.2. q-Permutation operator 39

5.3. q-Antisymmetrizer and quantum determinant 41

Chapter 6. Quantum Sylvester theorem for Uq(gln) 47

Chapter 7. Centraliser construction for Uq(gln) 53

Chapter 8. Skew representations of Y(gln) 63

8.1. Gelfand–Tsetlin characters 68

Chapter 9. Skew representations of Uq(gln) 71

9.1. Gelfand–Tsetlin characters 77

Bibliography 81

ix

CHAPTER 1

Definitions and preliminaries

In this chapter we define and give some basic properties of the algebras we shall be

dealing with, that is the enveloping algebra U(gln) of gln, the Yangian Y(gln), the quantized

enveloping algebra Uq(gln) and the quantum affine algebra Uq(gln). We also introduce the

quasi-determinants of a non-commutative algebra.

1.1. General definitions

If X =∑n

ij=1 xij ⊗ eij is an n × n matrix with entries xij taken from an algebra A,

where eij ∈ EndCn are the standard matrix units, then we denote by Xi the matrix X

acting in the ith copy of Cn:

Xi =N∑

i,j=1

xij ⊗ 1⊗(a−1) ⊗ eij ⊗ 1

⊗(m−a), ∈ A⊗ (EndCn)⊗m. (1.1)

Informally speaking, X may be considered as an “operator” on Cn with “coefficients” in

the algebra A. If e1, . . . , en are the standard basis vectors of Cn, then Xej is interpreted

as the linear combination

Xej =n∑

i=1

xij ⊗ ei ∈ A⊗ Cn.

If we have an operator

C =n∑

i,j,k,l=1

cijkl eij ⊗ ekl ∈ EndCn ⊗ EndCn,

then for any two indices a, b ∈ {1, . . . ,m} such that a < b, we define the element Cab of

the algebra (EndCn)⊗m by

Cab =n∑

i,j,k,l=1

cijkl 1⊗(a−1) ⊗ eij ⊗ 1

⊗(b−a−1) ⊗ ekl ⊗ 1⊗(m−b). (1.2)

Here the tensor factors eij and ekl belong to the a-th and b-th copies of EndCn, respectively.

Where appropriate, we will identify the element Cab with the element 1⊗Cab of the algebraA⊗ (EndCn)m.

1

2 1. DEFINITIONS AND PRELIMINARIES

1.2. The enveloping algebra U(gln)

The enveloping algebra U(gln) is as usual the associative algebra generated by Eij ,

1 6 i, j 6 n subject to

[Eij, Ekl] = δkjEil − δilEkj . (1.3)

To clarify the parallel between the ordinary and q-cases, we note that we can rewrite these

relations in matrix form. Put E(u) = I + Eu−1 and R(u) = I − Pu−1.

Proposition 1.1. The defining relations of U(gln) are equivalent to the ternary relation

R(u− v)E1(u)E2(v) = E2(v)E1(u)R(u− v) (1.4)

Proof. Expanding (1.4) and rearranging we get

[eij(u), ekl(v)] =1

u− v(ekj(u)eil(v)− ekj(v)eil(u)),

where eij(u) = δij + u−1Eij is the ijth entry of E(u). Simplifying gives [Eij , Ekl] =

δkjEil − Ekjδil. �

The classical Poincare–Birkhoff–Witt theorem provides an explicit basis for the en-

veloping algebra of a Lie algebra, for see for example [Dix], section 2.1:

Theorem 1.2. Let g be a Lie algebra, and x1, . . . , xn an ordered basis for the vector

space g. Then the elements xν11 , . . . , xνnn , where ν = (ν1, . . . , νn) ∈ N

n form a basis for the

enveloping algebra U(g).

We also state the classical Harish-Chandra isomorphism for comparison with analogues

for the other algebras we will study. Let U−, U+ and U0 denote the subalgebras of U(gln)

respectively generated by the Eij with i > j, the Eij with i < j and the Eii. By the

Poincare–Birkhoff–Witt theorem multiplication defines an isomorphism of vector spaces

U− ⊗ U0 ⊗ U+ ∼= U(gln). (1.5)

The products Em111 . . . E

mnnn with mi ∈ Z form a basis of U0. Moreover, the monomials

Ekn,n−1n,n−1 . . . Ekn2

n2 . . . Ek3232 E

kn1n1 . . . Ek21

21

with non-negative powers kij form a basis of U−, while the monomials

E k1212 . . . E k1n

1n E k2323 . . . E k2n

2n . . . Ekn−1,nn−1,n

with non-negative powers kij form a basis of U+.

Denote by Z the center of the algebra U(gln). Due to the isomorphism (1.34), any

z ∈ Z can be regarded as an element of U−⊗U0⊗U+. Denote by χ(z) the projection of zto the subalgebra U0 so that χ(z)− z belongs to the left ideal of U(gln) generated by theelements Eij with i < j. The map

χ : Z→ U0 (1.6)

1.3. THE YANGIAN Y(gln) 3

is an algebra homomorphism called the Harish-Chandra homomorphism. This homomor-

phism is injective and its image is the subalgebra of U0 generated by the symmetric poly-

nomials in E11, E22 − 1 . . . , Enn − n+ 1.

1.3. The Yangian Y(gln)

Definition 1.3. The Yangian for gln, which we denote Y(gln), is the unital associative

algebra over C with countably many generators t(1)ij , t(2)ij , . . . where i, j = 1, . . . , n, and the

defining relations

[t(r+1)ij , t

(s)kl ]− [t

(r)ij , t

(s+1)kl ] = t

(r)kj t(s)il − t

(s)kj t(r)il , (1.7)

where r, s = 0, 1, . . . and t(0)ij = δij. �

Introducing the formal generating series

tij(u) = δij + t(1)ij u

−1 + t(2)ij u

−2 + ∙ ∙ ∙ ∈ Y(gln)[[u−1]], (1.8)

we can write (1.7) in the form

(u− v) [tij(u), tkl(v)] = tkj(u) til(v)− tkj(v) til(u); (1.9)

the indeterminates u and v are considered to commute with each other and with the

elements of the Yangian.

The following is an equivalent form of the defining relations of the algebra Y(gln).

Proposition 1.4. The system of relations (1.7) is equivalent to the system

[t(r)ij , t

(s)kl ] =

min(r,s)∑

a=1

(t(a−1)kj t

(r+s−a)il − t(r+s−a)kj t

(a−1)il

). (1.10)

Proof. Observe that the multiplication of both sides of (1.9) by the formal series∑∞p=0 u

−p−1vp yields an equivalent relation

[tij(u), tkl(v)] =(tkj(u)til(v)− tkj(v)til(u)

) ∞∑

p=0

u−p−1vp.

Taking the coefficients of u−rv−s on both sides gives

[t(r)ij , t

(s)kl ] =

r∑

a=1

(t(a−1)kj t


(a−1)il

).

This agrees with (1.10) in the case r 6 s. Finally, if r > s observe that

r∑

a=s+1

(t(a−1)kj t


(a−1)il

)= 0

completing the proof. �


Introduce the n × n matrix T (u) ∈ Y(gln)[[u−1]] ⊗ EndCn whose ij-th entry is the

series tij(u), i.e.

T (u) =n∑

i,j=1

tij(u)⊗ eij , (1.11)

and consider the permutation operator

P =n∑

i,j=1

eij ⊗ eji ∈ EndCn ⊗ EndCn. (1.12)

The rational function

R(u) = 1− Pu−1 (1.13)

with values in EndCn ⊗ EndCn is called the Yang R-matrix , where we write 1 instead of

1⊗ 1, for brevity.

Proposition 1.5. In the algebra (EndCn)⊗3(u, v) we have the identity

R12(u)R13(u+ v)R23(v) = R23(v)R13(u+ v)R12(u). (1.14)

Proof. Multiplying both sides of the relation (1.14) by uv(u + v) we come to verify

the identity

(u+ P12)(u+ v + P13)(v + P23) = (v + P23)(u+ v + P13)(u+ P12). (1.15)

Each operator Pij is the image of the corresponding transposition (i j) of the symmet-

ric group S3 under the natural action on (CN)⊗3 by permutations of the tensor factors.

Therefore, (1.15) is immediate from the relations in the group algebra C [S3]. �

The relation (1.14) is known as the Yang–Baxter equation (with spectral parameters).

The Yang R-matrix is its simplest nontrivial solution. Below we regard T1(u) and T2(v) as

formal power series with coefficients from the algebra Y(gln) ⊗ End(Cn) ⊗ End(Cn). We

also identify R(u− v), and the rational function 1⊗R(u− v) taking values in this algebra.

Proposition 1.6. The defining relations (1.7) of the algebra Y(gln) can be written in

the equivalent form (the ‘ternary relation’)

R(u− v)T1(u)T2(v) = T2(v)T1(u)R(u− v). (1.16)

Proof. Let us apply the “operators” on both sides of the relation (1.16) to an arbitrary

basis vector ej ⊗ el ∈ Cn ⊗ Cn. For the left hand side we get∑

i,k

tij(u) tkl(v)⊗ ei ⊗ ek −1

u− v

∑

i,k

tij(u) tkl(v)⊗ ek ⊗ ei,

while the right hand side gives∑

i,k

tkl(v) tij(u)⊗ ei ⊗ ek −1

u− v

∑

i,k

tkj(v) til(u)⊗ ei ⊗ ek.

1.3. THE YANGIAN Y(gln) 5

Multiplying by u− v and equating the coefficients of ei ⊗ ek we recover (1.9). �

We shall often use formal series to define or describe maps between various algebras.

If a(u) and b(u) are formal power series in u−1 with coefficients in certain algebras then

assignments of the type a(u) 7→ b(u) are then understood in the sense that every coefficient

of a(u) is mapped to the corresponding coefficient of b(u).

Many applications of the Yangian are based on the following simple observation.

Proposition 1.7. The assignment

ev : tij(u) 7→ δij + Eiju−1 (1.17)

defines a surjective homomorphism Y(gln)→ U(gln). Moreover, the assignment

Eij 7→ t(1)ij (1.18)

defines an embedding U(gln) ↪→ Y(gln).

Proof. By Definition 1.3, we need to verify the equality

(u− v)[Eij, Ekl] u−1v−1

= (δkj + Ekju−1)(δil + Eilv

−1)− (δkj + Ekjv−1)(δil + Eilu

−1).

But this clearly holds due to the commutation relations in gln, which proves the first part

of the proposition. In order to prove the second part, put r = s = 1 in (1.10) which gives

[t(1)ij , t

(1)kl ] = δkjt

(1)il − δilt

(1)kj .

Thus, (1.18) is an algebra homomorphism which is clearly surjective. Its injectivity follows

from the observation that by applying (1.18) and then (1.17), we get the identity map on

U(glN). �

The homomorphism ev is called the evaluation homomorphism. By means of this ho-

momorphism any representation of the Lie algebra gln can be regarded as a representation

of Y(gln). Any irreducible representation of gln remains irreducible over Y(gln), by sur-

jectivity of ev.

Proposition 1.8. Each of the mappings

T (u) 7→ f(u)T (u), (1.19)

T (u) 7→ T (u− c), (1.20)

T (u) 7→ B T (u)B−1 (1.21)

defines an automorphism of Y(glN), where f(u) is a formal series in u−1 with constant

term 1, c is a complex constant, and B is an invertible n× n complex matrix.


Proof. Multiplying the ternary relation by f(u)f(v) proves (1.19). The shift u 7→ u−cis a well-defined operation in Y(gln)[[u

−1]] so (1.20) is clear since the ternary relation is

invariant under this shift of parameter. Conjugating the ternary relation by B1B2 and

observing that R(u− v)B1B2 = B2B1R(u− v) (because PB1P = B2) gives (1.21). �

Proposition 1.9. The mapping

ω = ωn : T (u) 7→ T−1(−u) (1.22)

defines an involutive automorphism of Y(gln).

Note that T−1(−u) is not equal to (T (−u))−1.

Proof. Multiply both sides of the ternary relation (1.16) on the right by T−12 T−11 and

on the left by T−11 T−12 . Conjugate by P and replace u and v by −v and −u respectively.This yields

R(u− v)T−11 (−u)T−12 (−v) = T

−12 (−v)T

−11 (−u)R(u− v).

To show that ω is an automorphism, apply ω to ω(T (u))T (−u) = 1.

ω2(T (u))ω(T (−u)) = 1,

ω2(T (u))T−1(u) = 1,

ω2(T (u)) = T (u).

That is, ω2 = id. �

Next note that a version of the classical Poincare–Birkhoff-Witt theorem holds for

Y(gln):

Theorem 1.10. If we linearly order the generators t(r)ij , then any element of Y(gln) can

be uniquely written as a linear combination of ordered monomials in the generators.

A proof can be found in [MNO], and we will provide a new proof for this theorem in

Chapter 4.

The Yangian has two natural filtrations, defined by setting deg t(r)ij = r and deg′ t

(r)ij =

r − 1. Denote the corresponding graded algebras by grY(gln) and gr′Y(gln). It is clear

from the defining relations that grY(gln) is commutative.

1.4. The quantum enveloping algebra Uq(gln)

We shall use an R-matrix presentation of the algebra Uq(gln) following [J1]; see also

[KS] for more details. We fix a complex parameter q which is nonzero and not a root of

unity. Consider the R-matrix

R = q∑

i

Eii ⊗ Eii +∑

i 6=j

Eii ⊗ Ejj + (q − q−1)∑

i<j

Eij ⊗ Eji (1.23)

1.4. THE QUANTUM ENVELOPING ALGEBRA Uq(glN ) 7

which is an element of EndCn ⊗ EndCn, where the Eij denote the standard matrix units

and the indices run over the set {1, . . . , n}. The R-matrix satisfies the Yang–Baxter equa-tion

R12R13R23 = R23R13R12, (1.24)

where both sides take values in EndCn ⊗ EndCn ⊗ EndCn and the indices indicate the

copies of EndCn, e.g., R12 = R⊗ 1 etc.The quantized enveloping algebra Uq(gln) is generated by elements tij and tij with

1 6 i, j 6 n subject to the relations

tij = tji = 0, 1 6 i < j 6 n,

tii tii = tii tii = 1, 1 6 i 6 n,

R T1T2 = T2T1R, RT 1T 2 = T 2T 1R, RT 1T2 = T2T 1R.

(1.25)

Here T and T are the matrices

T =∑

i,j

tij ⊗ Eij, T =∑

i,j

tij ⊗ Eij, (1.26)

which are regarded as elements of the algebra Uq(gln)⊗EndCn. Both sides of each of the

R-matrix relations in (1.25) are elements of Uq(gln)⊗EndCn⊗EndCn and the indices of

T and T indicate the copies of EndCn where T or T acts; e.g. T1 = T ⊗ 1.Note that we can rewrite the defining RTT relations in terms of the alternative R-

matrix

R = q−1∑

i

Eii ⊗ Eii +∑

i 6=j

Eii ⊗ Ejj − (q − q−1)∑

i<j

Eij ⊗ Eji (1.27)

as

R T1T2 = T2T1R, R T 1T 2 = T 2T 1R, R T1T 2 = T 2T1R. (1.28)

In terms of the generators the defining relations between the tij can be written as

qδij tia tjb − qδab tjb tia = (q − q

−1) (δb<a − δi<j) tja tib, (1.29)

where δi<j equals 1 if i < j and 0 otherwise. The relations between the tij are obtained

by replacing tij by tij everywhere in (1.29). Finally, the relations involving both tij and tijhave the form

qδij tia tjb − qδab tjb tia = (q − q

−1) (δb<a tja tib − δi<j tja tib). (1.30)

It is well known that the algebra Uq(gln) specializes to U(gln) as q → 1. To make thismore precise, regard q as a formal variable and Uq(gln) as an algebra over C(q). Then setA = C [q, q−1] and consider the A-subalgebra UA of Uq(gln) generated by the elements

tij

q − q−1for i > j,

tij

q − q−1for i < j, (1.31)


andtii − 1q − 1

,tii − 1q − 1

, (1.32)

for i = 1, . . . , n. Then we have an isomorphism

UA ⊗A C ∼= U(gln) (1.33)

with the action of A on C defined via the evaluation q = 1; see e.g. [CP, Section 9.2].Note that the elements (1.31) respectively specialize to the elements Eij and −Eij of U(gln)while the elements (1.32) specialize to Eii and −Eii.The quantized enveloping algebra Uq(sln) is defined as the associative algebra with

generators k1, . . . , kn−1, k−11 , . . . , k−1n−1, e1, . . . , en−1 and f1, . . . , fn−1 subject to the defining

relationskikj = kj ki, kik

−1i = k

−1i ki = 1,

kiej k−1i = q

aij ej, kifj k−1i = q

−aijfj,

[ei, fj] = δijki − k

−1i

q − q−1,

[ei, ej] = [fi, fj] = 0 if |i− j| > 1,

e2i ej − (q + q−1)eiej ei + ej e

2i = 0 if |i− j| = 1,

f 2i fj − (q + q−1)fifj fi + fj f

2i = 0 if |i− j| = 1,

where [aij] denotes the Cartan matrix associated with the Lie algebra sln so that its only

nonzero entries are aii = 2 and aij = −1 for |i− j| = 1.We have an embedding Uq(sln) ↪→ Uq(gln) given by

ki 7→ tii ti+1,i+1, k−1i 7→ tii ti+1,i+1, ei 7→ −ti,i+1 tii

q − q−1, fi 7→

tii ti+1,i

q − q−1.

We shall identify Uq(sln) with a subalgebra of Uq(gln) via this embedding.

An analogy of the Poincare–Birkhoff–Witt theorem holds for Uq(gln). Let U−q , U

+q

and U0q denote the subalgebras of Uq(gln) respectively generated by the tij with i > j,

the tij with i < j and the tii, tii with all i. It is implied by [CP, Proposition 9.2.2] that

multiplication defines an isomorphism of vector spaces

U−q ⊗ U0q ⊗ U

+q∼= Uq(gln). (1.34)

The products tm111 . . . tmnnn with mi ∈ Z form a basis of U0q. Moreover, the monomials

tkn,n−1n,n−1 . . . t

kn2n2 . . . t

k3232 t

kn1n1 . . . tk2121

with non-negative powers kij form a basis of U−q , while the monomials

t k1212 . . . t k1n1n t k2323 . . . t k2n2n . . . tkn−1,nn−1,n

with non-negative powers kij form a basis of U+q .

1.5. QUANTUM AFFINE ALGEBRA 9

Denote by Zq the center of the algebra Uq(gln). Due to the isomorphism (1.34), any

z ∈ Zq can be regarded as an element of U−q ⊗U0q ⊗U

+q . Denote by χ(z) the projection of

z to the subalgebra U0q so that χ(z) − z belongs to the left ideal of Uq(gln) generated bythe elements tij with i < j. The map

χ : Zq → U0q (1.35)

is an algebra homomorphism called the Harish-Chandra homomorphism. This homomor-

phism is injective and its image is the subalgebra of U0q generated by the symmetric poly-

nomials in x21, . . . , x2n and the polynomial x

−11 . . . x−1n , where xi = q

−i+1 tii. Moreover, those

polynomials of total degree zero in x1, . . . , xn form the image of the center of the subalgebra

Uq(sln) under the Harish-Chandra homomorphism; see e.g. [CP, Proposition 9.2.5] and

[KS, Section 6.3.4].

1.5. Quantum affine algebra

Now consider the Lie algebra of Laurent polynomials gln[λ, λ−1] in an indeterminate λ.

We denote it by gln for brevity. The quantum affine algebra Uq(gln) is a deformation of the

universal enveloping algebra U(gln). By definition, Uq(gln) has countably many generators

t(r)ij and t

(r)ij where 1 6 i, j 6 n and r runs over nonnegative integers. They are combined

into the matrices

T (u) =n∑

i,j=1

tij(u)⊗ Eij, T (u) =n∑

i,j=1

tij(u)⊗ Eij , (1.36)

where tij(u) and tij(u) are formal series in u−1 and u, respectively:

tij(u) =∞∑

r=0

t(r)ij u

−r, tij(u) =∞∑

r=0

t(r)ij ur. (1.37)

The defining relations are

t(0)ij = t

(0)ji = 0, 1 6 i < j 6 n,

t(0)ii t

(0)ii = t

(0)ii t

(0)ii = 1, 1 6 i 6 n,

R(u, v)T1(u)T2(v) = T2(v)T1(u)R(u, v),

R(u, v)T 1(u)T 2(v) = T 2(v)T 1(u)R(u, v),

R(u, v)T 1(u)T2(v) = T2(v)T 1(u)R(u, v),

(1.38)


where we have used the notation of (1.25) and R(u, v) = Rq(u, v) is the trigonometric

R-matrix given by

Rq(u, v) = (u− v)∑

i 6=j

Eii ⊗ Ejj + (q−1u− qv)

∑

i

Eii ⊗ Eii

+ (q−1 − q)u∑

i>j

Eij ⊗ Eji + (q−1 − q)v

∑

i<j

Eij ⊗ Eji.(1.39)

It satisfies the Yang–Baxter equation (with spectral parameters)

R12(u, v)R13(u,w)R23(v, w) = R23(v, w)R13(u,w)R12(u, v), (1.40)

where both sides take values in EndCn ⊗ EndCn ⊗ EndCn and the indices indicate the

copies of EndCn, e.g., R12(u, v) = R(u, v) ⊗ 1 etc.; see e.g. [DF], [FM] for more detailson the structure of Uq(gln).

In terms of generators the defining relations of Uq(gln) are given by

(q−δiku− q−δikv) tij(u)tkl(v) + (q−1 − q)(δi>ku+ δi<kv) tkj(u)til(v)

= (q−δjlu− q−δjlv) tkl(v)tij(u) + (q−1 − q)(δj>lv + δj<lu) tkj(v)til(u),

(1.41)


= (q−δjlu− q−δjlv) tkl(v)tij(u) + (q−1 − q)(δj>lv + δj<lu) tkj(v)til(u),

(1.42)


= (q−δjlu− q−δjlv) tkl(v)tij(u) + (q−1 − q)(δj>lv + δj<lu) tkj(v)til(u).

(1.43)

The subalgebra of Uq(gln) generated by the elements t(r)ij and t

(0)ii was studied e.g. in

[C], [NT1], [RTF]. We call it the q-Yangian and denote by Yq(gln).

The quantized enveloping algebra Uq(gln) is a natural subalgebra of Uq(gln) defined by

the embedding

tij 7→ t(0)ij , tij 7→ t

(0)ij . (1.44)

The fact that this is indeed an embedding will follow from the Poincare–Birkhoff–Witt

theorem for Uq(gln) (Corollary (7.5)).

Lemma 1.11. The function ev : Uq(gln)→ Uq(gln)

T (u) 7→ T − T u−1, T (u) 7→ T − T u (1.45)

is an algebra homomorphism which we call the evaluation homomorphism.

1.6. QUASI-DETERMINANTS 11

Proof. Clearly ev(t(0)ij ) = ev(t

(0)ji ) = 0, and ev(t

(0)ii ) ev(t

(0)ii ) = ev(t

(0)ii ) ev(t

(0)ii ) = 1 for

1 6 i < j 6 n. We need to show that T ′(u) := T − Tu−1, T′(u) := T − Tu satisfy

R(u, v)T ′1(u)T′2(v) = T

′2(v)T

′1(u)R(u, v),

R(u, v)T′1(u)T

′2(v) = T

′2(v)T

′1(u)R(u, v),

and R(u, v)T′1(u)T

′2(v) = T

′2(v)T

′1(u)R(u, v).

(1.46)

We have R(u, v)T ′1(u)T′2(v)− T

′2(v)T

′1(u)R(u, v)

=[(u− v)R + (q−1 − q)uP

](T1T2 + T 1T 2u

−1v−1 − T1T 2v−1 − T 1T2u

−1)

− (T2T1 + T 2T 1u−1v−1 − T 2T1v

−1 − T2T 1u−1)[(u− v)R + (q−1 − q)uP

]

= (u− v)((RT1T2 − T2T1R) + (RT 1T 2 − T 2T 1R)u−1v−1

− (RT 1T2 − T2T 1R)u−1 − (RT1T 2 − T 2T1R)v

−1)

+ (q−1 − q)u((PT1T2 − T2T1P ) + (PT 1T 2 − T 2T 1P )u−1v−1

− (PT 1T2 − T2T 1P )u−1 − (PT1T 2 − T 2T1P )v

−1)

= (1− uv−1)(RT1T 2 − T 2T1R) = 0,(1.47)

where in the last line we use (1.28). The other two relations follow from the fact that

T′(u) = −uT ′(u). �

Note that the evaluation is the identity on the subalgebra Uq(gln).

Lemma 1.12. For any nonzero constant α, the mapping

T (u) 7→ T (αu), T (u) 7→ T (αu) (1.48)

defines an automorphism of the algebra Uq(gln).

Proof. This is immediate since the R-matrix satisfies R(αu, αv) = αR(u, v). �

1.6. Quasi-determinants

Quasi-determinants were introduced by Gelfand and Retakh [GR] to fill the role for

non-commutative algebras which the determinant plays in the theory of commutative al-

gebras.

Let A = [aij ] be an n×n matrix over a ring with 1. Denote by Aij the matrix obtainedfrom A by deleting the i-th row and j-th column. Suppose that the matrix Aij is invertible.

Definition 1.13. The ij-th quasideterminant of A is defined by the formula

|A|ij = aij −∑

k 6=j,l 6=i

aik((Aij)−1

)klalj.


In a more graphic fashion, the quasideterminant |A|ij is denoted by boxing the entryaij:

|A|ij =

∣∣∣∣∣∣∣∣∣∣∣

a11 . . . a1j . . . a1N. . . . . .

ai1 . . . aij . . . aiN

. . . . . .

aN1 . . . aNj . . . aNN

∣∣∣∣∣∣∣∣∣∣∣

.

Example 1.14. For a 2× 2 matrix A the four quasideterminants are

|A|11 = a11 − a12 a−122 a21, |A|12 = a12 − a11 a

−121 a22,

|A|21 = a21 − a22 a−112 a11, |A|22 = a22 − a21 a

−111 a12. �

Lemma 1.15. Suppose that the submatrix A11 of the matrix A is invertible. Then the

system

a11x1+ ∙ ∙ ∙+ a1NxN = y1

a21x1+ ∙ ∙ ∙+ a2NxN = 0

. . .

aN1x1+ ∙ ∙ ∙+ aNNxN = 0

implies that y1 = |A|11 x1.

Proof. The column vector with coordinates x2, . . . , xN can be expressed from the last

N − 1 equations as

x2...

xN

= −

(A11)−1

a21...

aN1

x1.

Substituting this into the first equation we get

(a11 −

N∑

k, l=2

a1k((A11)−1

)klal1

)x1 = y1.

By Definition 1.13, this is the desired relation. �

Proposition 1.16. Suppose that the inverse matrix A−1 exists and that its ji-th entry

(A−1)ji is an invertible element of the ring. Then the ij-th quasideterminant of A is defined

and given by

|A|ij =((A−1)ji

)−1.

1.6. QUASI-DETERMINANTS 13

Proof. First consider the case i = j = 1. Let B = A−1 so that the element b11 is

invertible. Using block multiplication of matrices we derive from AB = 1 that

A11

(

B11 −

b21...

bN1

b−111 [b12 . . . b1N ]

)

= 1

proving that A11 is invertible so that |A|11 is defined. Now let b be the first column of B.Then Ab = e1, where e1 is the column vector which has 1 on the first position and zeros

elsewhere. Lemma 1.15 gives |A|11 b11 = 1 proving the claim. In the case of arbitrary i andj we rearrange the matrix A as follows: move the i-th row up to the top position and then

move the j-th column to the leftmost position. Rearranging the matrix B accordingly, we

reduce the argument to the previous case which gives |A|ij bji = 1. �

In particular, if the entries of the matrix A belong to a commutative ring then the ij-th

quasideterminant of A can be given by

|A|ij = (−1)i+j detA

detAij. (1.49)

CHAPTER 2

Quantum determinant of Y(gln)

We recall the well-known construction of the quantum determinant for the algebra

Y(gln). For any r > 2 introduce the rational function R(u1, . . . , ur) in r independentcomplex variables u1, . . . , ur, with values in the tensor product algebra (EndCn)⊗r, by

R(u1, . . . , ur) =∏

i<j

Rij(ui, uj). (2.1)

The product here taken in the lexicographical order on the pairs (i, j). That is,

R(u1, . . . , ur) := (Rr−1,r)(Rr−2,rRr−2,r−1) ∙ ∙ ∙ (R1r ∙ ∙ ∙R12), (2.2)

where we abbreviate Rij(ui, uj) by Rij.

Applying the Yang–Baxter equation (1.14) and using the obvious fact that Rij and Rkl

commute if the indices i, j, k, l are distinct, we can write (2.2) in several different forms.

In particular, a simple induction argument shows that we have

R(u1, . . . , um) = (R12 . . . R1m) ∙ ∙ ∙ (Rm−2,m−1Rm−2,m)(Rm−1,m). (2.3)

Below we will use the formal power series T1(u1), . . . , Tm(um) with coefficients from the

algebra Y(gln) ⊗ (EndCn)⊗m. We will identify R(u1, . . . , um) with the rational function

R(u1, . . . , um)⊗ 1 taking values in that algebra.Then we have the following corollary of the Yang–Baxter equation (1.14) and the

ternary relation (1.16):

Proposition 2.1 (Fundamental identity).

R(u1, . . . , ur)T1(u1) ∙ ∙ ∙Tr(ur) = Tr(ur) ∙ ∙ ∙T1(u1)R(u1, . . . , ur) (2.4)

Proof. To simplify the notation, set Ta = Ta(ua). We first note that

(R1r ∙ ∙ ∙R12)T1(T2 ∙ ∙ ∙Tr) = (T2 ∙ ∙ ∙Tr)T1(R1r ∙ ∙ ∙R12). (2.5)

Indeed,

(R1r ∙ ∙ ∙R12)T1(T2 ∙ ∙ ∙Tr) = R1r ∙ ∙ ∙R13(R12T1T2)T3 ∙ ∙ ∙Tr)

= R1r ∙ ∙ ∙R13(T2T1R12)T3 ∙ ∙ ∙Tr)

= T2(R1r ∙ ∙ ∙R13)T1(T3 ∙ ∙ ∙Tr)R12,

(2.6)

15

16 2. QUANTUM DETERMINANT OF Y(gln)

where we use first the ternary relation (1.16) and then the fact that Rij and Tk commute

if k is equal to neither i nor j. Repeating this procedure we interchange T1 successively

with T3, . . . , Tr and arrive at (2.5). Next we see that

R(u1, . . . , ur) = R(u2, . . . , ur)(R1r ∙ ∙ ∙R12). (2.7)

Using (2.5) we interchange T1 with T2 ∙ ∙ ∙Tr:

R(u1, . . . , ur)T1 ∙ ∙ ∙Tr = R(u2, . . . , ur)(R1r ∙ ∙ ∙R12)T1(T2 ∙ ∙ ∙Tr)

= R(u2, . . . , ur)(T2 ∙ ∙ ∙Tr)T1(R1r ∙ ∙ ∙R12).(2.8)

We continue by interchanging T2 with T3 ∙ ∙ ∙Tr. Repeating this procedure we arrive at theright hand side of (2.4). �

Let Sr denote the symmetric group acting on the set {1, . . . , r} . Consider the anti-symmetrizer in the group algebra of Sr,

∑

σ∈Sr

sgn σ ∙ σ ∈ C [Sr],

where sgn σ = (−1)`(σ) is the sign of σ. Denote by Ar the image of this anti-symmetrizerunder the natural action of Sr on the tensor space (Cn)⊗r, that is

Ar =∑

σ∈Sr

sgn σ ∙ Pσ ∈ (EndCn)⊗r. (2.9)

Keeping the notation e1, . . . , en for the standard basis vectors of Cn we thus have

Am(ei1 ⊗ ∙ ∙ ∙ ⊗ eir) =∑

σ∈Sr

sgn σ ∙ eiσ(1) ⊗ ∙ ∙ ∙ ⊗ eiσ(r) .

Note that this operator satisfies the relation AmAn = AnAm = m!An for m 6 n.

Proposition 2.2. If ui − ui+1 = 1 for all i = 1, . . . ,m− 1 then

R(u1, . . . , um) = Am.

Proof. We use induction on m. Case m = 2 is obvious for

A2 = R12(1) = 1− P12.

Due to (2.7), by the induction hypothesis we have for m > 2

R(u1, . . . , um) = A′m−1 (R1m . . . R12),

where A′m−1 denotes the anti-symmetrizer corresponding to the subset of indices {2, . . . ,m}.Observe that for any indices m > i1 > i2 > ∙ ∙ ∙ > ik > 1 we have

A′m−1 P1i1 P1i2 . . . P1ik = A′m−1 Pi1i2 . . . Pi1ik P1i1 = (−1)

k−1A′m−1 P1i1 . (2.10)

2. QUANTUM DETERMINANT OF Y(gln) 17

Therefore,

A′m−1 (R1m . . . R12) = A′m−1

(

1−P1m

m− 1

)

. . .

(

1−P12

1

)

= A′m−1 (1− α2 P12 − ∙ ∙ ∙ − αm P1m)(2.11)

with

αr =1

r − 1

∑

k>1

∑

r>i2>∙∙∙>ik>1

1

(i2 − 1) ∙ ∙ ∙ (ik − 1)= 1

for any r = 2, . . . ,m. It remains to note that

A′m−1 (1− P12 − ∙ ∙ ∙ − P1m) = Am. �

By Propositions 2.1 and 2.2 we have the following equality of formal power series in

u−1 with coefficients in Y(gln)⊗ (EndCn)⊗m,

Am T1(u) ∙ ∙ ∙Tm(u−m+ 1) = Tm(u−m+ 1) ∙ ∙ ∙T1(u)Am. (2.12)

Here we identify Am with Am ⊗ 1. Now take m = n. The operator An on (Cn)⊗n is

one-dimensional. Therefore, the element (2.12) with m = n equals An times a scalar series

with coefficients in Y(gln). This prompts the following definition.

Definition 2.3. The quantum determinant of the matrix T (u) with the coefficients in

Y(gln) is the formal series

qdetT (u) = 1 + d1u−1 + d2u

−2 + . . .

such that the element (2.12) with m = n, equals An qdetT (u).

Proposition 2.4. For any permutation q ∈ Sn we have

qdetT (u) = sgn q∑

p∈Sn

sgn p ∙ tp(1),q(1)(u) . . . tp(n),q(n)(u− n+ 1) (2.13)

= sgn q∑

p∈Sn

sgn p ∙ tq(1),p(1)(u− n+ 1) . . . tq(n),p(n)(u). (2.14)

In particular,

qdetT (u) =∑

p∈Sn

sgn p ∙ tp(1),1(u) . . . tp(n),n(u− n+ 1) (2.15)

=∑

p∈Sn

sgn p ∙ t1,p(1)(u− n+ 1) . . . tn,p(n)(u). (2.16)

Proof. Let us apply both sides of the identity

An T1(u) . . . Tn(u− n+ 1) = An qdetT (u)


to the vector eq(1) ⊗ . . .⊗ eq(n). By applying the left hand side, we get∑

i1,...,in

An(ei1 ⊗ . . .⊗ ein)⊗ ti1,q(1)(u) . . . tin,q(n)(u− n+ 1).

The vector An(ei1⊗. . .⊗ein) is zero unless the sequence of indices i1, . . . , in is a permutationp(1), . . . , p(n) of the sequence 1, . . . , n for some p ∈ Sn. Then

An(ep(1) ⊗ . . .⊗ ep(n)

)= sgn p ∙ An(e1 ⊗ . . .⊗ en).

The application of the right hand side gives

sgn q ∙ qdetT (u) ∙ An(e1 ⊗ . . .⊗ en),

which completes the proof of (2.13). For the proof of (2.14) we use the identity

Tn(u− n+ 1) . . . T1(u)An = An qdetT (u).

Apply both sides to the vector en ⊗ . . . ⊗ e1 and compare the coefficients of the vector

eq(n) ⊗ . . .⊗ eq(1). �

Example 2.5. In the case n = 2 we have

qdetT (u) = t11(u) t22(u− 1)− t21(u) t12(u− 1)

= t22(u) t11(u− 1)− t12(u) t21(u− 1)

= t11(u− 1) t22(u)− t12(u− 1) t21(u)

= t22(u− 1) t11(u)− t21(u− 1) t12(u). �

More generally, assuming that m 6 n is arbitrary, we can define m×m quantum minorsas the matrix elements of the operator (2.12). Namely, the operator (2.12) can be written

as ∑t a1... amb1... bm

(u)⊗ ea1b1 ⊗ . . .⊗ eambm ,

summed over the indices ai, bi ∈ {1, . . . , n}, where ta1... amb1... bm

(u) ∈ Y(gln)[[u−1]]. We call these

elements the quantum minors of the matrix T (u). The following formulae are obvious

generalizations of (2.13) and (2.14),

t a1... amb1... bm(u) =

∑

p∈Sm

sgn p ∙ tap(1)b1(u) . . . tap(m)bm(u−m+ 1) (2.17)

=∑

p∈Sm

sgn p ∙ ta1bp(1)(u−m+ 1) . . . tambp(m)(u). (2.18)

It is clear from the definition that the quantum minors are skew-symmetric with respect

to permutations of the upper indices and of the lower indices:

tap(1)... ap(m)b1... bm

(u) = sgn p ∙ t a1... amb1... bm(u) and t a1... ambp(1)... bp(m)

(u) = sgn p ∙ t a1... amb1... bm(u)

for any p ∈ Sm. The following quantum analogues of the row and column expansionformulae take place.


Proposition 2.6. We have the relations

t a1... amb1... bm(u) =

m∑

l=1

(−1)m−l t a1...al... amb1... bm−1(u) talbm(u−m+ 1)

=m∑

l=1

(−1)m−l t a1... am−1b1...bl... bm

(u− 1) tambl(u)

=m∑

l=1

(−1)l−1 talb1(u) ta1...al... amb2... bm

(u− 1)

=m∑

l=1

(−1)l−1 ta1bl(u−m+ 1) ta2... am

b1...bl... bm(u).

Proof. All relations are immediate from the explicit formulae for the quantum minors.

�

In the following k, l, ai, bj are arbitrary indices from the set {1, . . . , n}.

Proposition 2.7. We have the relations

(u− v) [ tkl(u), ta1... amb1... bm

(v)] =m∑

i=1

tail(u) ta1... k ... amb1 ... bm

(v)−m∑

i=1

t a1 ... amb1... l ... bm

(v) tkbi(u)

and

(u− v +m− 1) [ tkl(u), ta1... amb1... bm

(v)]

=m∑

i=1

t a1... k ... amb1 ... bm(v) tail(u)−

m∑

i=1

tkbi(u) ta1 ... amb1... l ... bm

(v),

where the indices k and l in the quantum minors replace ai and bi, respectively.

Proof. By Proposition 2.1, we have the relation

R(u, v, v − 1, . . . , v −m+ 1)T0(u)T1(v) . . . Tm(v −m+ 1)

= Tm(v −m+ 1) . . . T1(v)T0(u)R(u, v, v − 1, . . . , v −m+ 1), (2.19)

where we have used an extra copy of the algebra EndCN labelled by 0. Using (2.2) and

Proposition 2.2 we get

R(u, v, v − 1, . . . , v −m+ 1) = AmR0m(u− v +m− 1) . . . R01(u− v).

Then we proceed exactly as in the proof of Proposition 2.2 by using the relation (2.10)

adjusted to our present notation. The calculation gives the following generalization of

(2.11):

R(u, v, v − 1, . . . , v −m+ 1) = Am

(

1−1

u− v(P01 + ∙ ∙ ∙+ P0m)

)

. (2.20)


Now apply both sides of (2.19) to the vector el⊗eb1⊗ . . .⊗ebm and compare the coefficientsof the vector ek⊗ea1⊗ . . .⊗eam . This yields the first relation. In order to prove the second,we use the relation

R(v, v − 1, . . . , v −m+ 1, u)T1(v) . . . Tm(v −m+ 1)Tm+1(u)

= Tm+1(u)Tm(v −m+ 1) . . . T1(v)R(v, v − 1, . . . , v −m+ 1, u)

implied by Proposition 2.1. Similar to the above argument, we show with the use of (2.3)

that

R(v, v − 1, . . . , v −m+ 1, u) = Am

(

1 +1

u− v +m− 1(P1,m+1 + ∙ ∙ ∙+ Pm,m+1)

)

.

The argument is now completed in the same way as for the first formula. �

Proposition 2.8. For any indices i, j we have

[taibj(u), ta1... amb1... bm

(v)] = 0. (2.21)

Proof. The quantum minor is zero if it has two repeated upper or lower indices.

Therefore, the right hand side of either relation of Proposition 2.7 with k = ai and l = bjwill be the commutator which occurs in the left hand side. Hence the commutator is

zero. �

We have as a consequence that the coefficients d1, d2, . . . of the quantum determinant

qdetT (u) belong to the center of Y(gln). In fact we will show they are algebraically

independent and generate the center.

We record the following relation derived in the proof of Proposition 2.7.

Corollary 2.9. We have

AmR0m(u− v +m− 1) . . . R01(u− v) = Am

(

1−1

u− v(P01 + ∙ ∙ ∙+ P0m)

)

.

For the proof of the next theorem we need a general property of the universal enveloping

algebras of the form U(g [z]), where z is an indeterminate and g [z] ∼= g ⊗ C [z] is thepolynomial current Lie algebra corresponding to a Lie algebra g. A proof can be found in,

for example, [MNO].

Lemma 2.10. Let a be a subalgebra of a finite-dimensional Lie algebra g such that a is

reductive in g. Let b be the centralizer of a in g. Then the centralizer of U(a [z]) in U(g [z])

is equal to U(b [z]). Moreover, if the center of g is trivial then the center of U(g [z]) is also

trivial.

We can now prove the following.


Theorem 2.11. The coefficients d1, d2, . . . of the series qdetT (u) belong to the center

of the Yangian Y(gln). Moreover, these elements are algebraically independent and generate

the center.

Proof. The first claim of the theorem follows from Proposition 2.8. To prove the

second claim, let us consider the filtration on the algebra Y(gln) defined by deg′ t(r)ij =

r−1. The corresponding graded algebra gr′Y(gln) is isomorphic to the universal envelopingalgebra U(glN [z]). To see this note that the images t

(r)ij of t

(r)ij in gr

′Y(gln) satisfy

[t(r)ij , t

(s)kl ] = δkjt

(r+s−1)il − δilt

(r+s−1)kj ,

so that Eijzr−1 7→ t

(r)ij defines a surjective algebra homomorphism U(gln[z]) → gr

′Y(gln).

Its kernel is trivial due to the Poincare–Birkhoff–Witt theorem. Now we derive from (2.15)

that the coefficient dr of qdetT (u) has the form

dr = t(r)11 + ∙ ∙ ∙+ t

(r)nn plus terms of degree less than r − 1.

By the Poincare–Birkhoff–Witt theorem, this implies that the elements dr with r > 1 arealgebraically independent. Furthermore, the image of the element dr in the (r − 1)-thcomponent of gr′Y(gln) coincides with I z

r−1 where I = E11 + ∙ ∙ ∙+Enn. By Lemma 2.10,the center of U(sln[z]) is trivial. Therefore the elements I z

r−1 with r > 1 generate thecenter of U(gln[z]). �

CHAPTER 3

Quantum Sylvester theorem for Y(gln)

Suppose that A = [aij ] is a numerical (m + n) × (m + n) matrix. For any indices

i, j = 1, . . . , n introduce the minors cij of A corresponding to the rows 1, . . . , m,m+ i and

columns 1, . . . , m,m+ j so that

cij = a 1...m,m+i1...m,m+j .

Let A(m) be the submatrix of A determined by the first m rows and columns. The classical

Sylvester theorem (see for example [T] or [L]) provides a formula for the determinant of

the matrix C = [cij ]:

detC = detA ·(detA(m)

)n−1.

We shall give a Yangian analogue of this result in Theorem 3.5, where the minors of A

are replaced by quantum minors of the matrix T (u).

For any m > 0 introduce the homomorphism ϕm

ϕm : Y(gln)→ Y(glm+n) (3.1)

which takes tij(u) to tm+i,m+j(u). (In fact, it will follow from the Poincare–Birkhoff–Witt

theorem for Y(gln) (4.12) that this homomorphism is injective.) Consider the composition

ψm = ωm+n ◦ ϕm ◦ ωn, (3.2)

where ωn is the involutive automorphism of Y(gln) defined in Proposition 1.9. Then ψm is

an algebra homomorphism

ψm : Y(gln)→ Y(glm+n).

Its action on the generators of Y(gln) can be expressed in terms of quasideterminants (see

Definition 1.13) as follows.

Lemma 3.1. For any 1 6 i, j 6 n, we have

ψm : tij(u) 7→

∣∣∣∣∣∣∣∣∣

t11(u) . . . t1m(u) t1,m+j(u)...

. . ....

...

tm1(u) . . . tmm(u) tm,m+j(u)

tm+i,1(u) . . . tm+i,m(u) tm+i,m+j(u)

∣∣∣∣∣∣∣∣∣.

23

24 3. QUANTUM SYLVESTER THEOREM FOR Y(glN )

Proof. Introduce the sets of indices

P = {1, . . . ,m} and Q = {m+ 1, . . . ,m + n}.

Consider the block partitioned matrix[A(u) B(u)

C(u) D(u)

]

whose entries are the series tij(u) so that

A(u) = T (u)PP , B(u) = T (u)PQ, C(u) = T (u)QP and D(u) = T (u)QQ.

Let [A(u) B(u)

C(u) D(u)

]−1=

[A′(u) B′(u)

C ′(u) D′(u)

]

.

By definition, the homomorphism ψm maps ωn(T (u)) = T−1(−u) to D′(−u). Hence

ψm : T (u) 7→ D(u)− C(u)A(u)−1B(u).

Taking the ij-th entry and using Definition 1.13 we get the desired formula for the image

of tij(u). �

Due to the defining relations (1.25), there are natural homomorphisms

ιm : Y(glm)→ Y(glm+1)

t(r)ij 7→ t

(r)ij .

It will follow from the Poincare–Birkhoff–Witt theorem that ιm is injective. By Lemma 3.1,

the description of ψm(tij(u)) does not depend on n. Therefore, the maps ψm are compatible

with the homomorphisms ιm. In other words, the following diagram commutes

Y(gl1)ι1−−−→ Y(gl2)

ι2−−−→ Y(gl3)ι3−−−→ . . .

ψm

y ψm

y ψm

y

Y(glm+1) −−−→ιm+1

Y(glm+2) −−−→ιm+2

Y(glm+3) −−−→ιm+3

. . . .

Note also the property

ψl ◦ ψm = ψl+m (3.3)

implied by the definition of ψm.

We next describe the action of ψm on the quantum minors (see Chapter 2). Suppose

that m is an integer such that 0 6 m 6 n. For any two subsets

P = {i1, . . . , im} and Q = {j1, . . . , jm}

of {1, . . . , n} with cardinality m, let

P = {im+1, . . . , in} and Q = {jm+1, . . . , jn}

3. QUANTUM SYLVESTER THEOREM FOR Y(glN ) 25

be their set complements in {1, . . . , n}. We assume that

i1 < ∙ ∙ ∙ < im and j1 < ∙ ∙ ∙ < jm ,

im+1 < ∙ ∙ ∙ < in and jm+1 < ∙ ∙ ∙ < jn .

For any n × n matrix X, we will denote by XPQ the submatrix whose rows and columnsare numbered by the elements of the sets P and Q respectively.Each of the sequences i1, . . . , in and j1, . . . , jn above is a permutation of the sequence

1, . . . , n. Denote these two permutations by i and j respectively. If A is a n × n matrixwith complex entries and B is the inverse matrix, then the following classical identity for

complementary minors of A and B holds (see for example [T] or [L]):

detA ∙ bjm+1...jnim+1...in= sgn i ∙ sgn j ∙ a i1...imj1...jm

. (3.4)

Here a i1...imj1...jmis the minor of A corresponding to the submatrix APQ , and b

jm+1...jnim+1...in

is the

minor of B corresponding to the submatrix BQP .

We shall prove an analogue of the identity (3.4) for the matrix T (u). We first find the

action of the automorphism ω (defined by (1.9)) on the quantum minors of the matrix

T (u), introduced in Chapter 2.

Proposition 3.2. We have the identity

qdetT (u) ∙ ωn(tjm+1...jnim+1...in

(−u+ n− 1))= sgn p ∙ sgn q ∙ t i1...imj1...jm

(u) .

Proof. By Definition 2.3,

qdetT (u)An = An T1 . . . Tn, (3.5)

where Ti = Ti(u−i+1) for i = 1, . . . , n. Let us multiply both sides of (3.5) by T−1n . . . T−1m+1from the right. Then (3.5) takes the form

qdetT (u)An T−1n . . . T−1m+1 = An T1 . . . Tm . (3.6)

now we apply both sides of (3.6) to the basis vector

ej1 ⊗ . . .⊗ ejm ⊗ eim+1 ⊗ . . .⊗ ein ∈ (Cn)⊗n. (3.7)

By applying the right hand side of (3.6) to this vector, we get∑

a1,...,am

An(ea1 ⊗ . . .⊗ eam ⊗ eim+1 ⊗ . . .⊗ ein) ⊗ ta1j1(u) . . . tamjm(u−m+ 1) . (3.8)

The summation here can obviously be restricted to the sequences a1, . . . , am which are

permutations of the sequence i1, . . . , im . Consider the vector An(e1 ⊗ . . . ⊗ en). The sum(3.8) is proportional to this vector, with the coefficient

sgn p ∙ t i1...imj1...jm(u) .

26 3. QUANTUM SYLVESTER THEOREM FOR Y(glN )

We shall use the notation t ′ij(u) for the entries of the matrix T−1(u). Then, by applying

the left hand side of (3.6) to the basis vector (3.7), we obtain

∑

bm+1,...,bn

An(ej1 ⊗ ∙ ∙ ∙ ⊗ ejm ⊗ ebm+1 ⊗ ∙ ∙ ∙ ⊗ ebn)

⊗ t ′bnin(u− n+ 1) ∙ ∙ ∙ t′bm+1im+1

(u−m) .

By (2.13), here the coefficient of An(e1 ⊗ ∙ ∙ ∙ ⊗ en) equals

sgn q ∙ tjm+1...jnim+1...in(−u+ n− 1) .

Thus Proposition 3.2 follows from the equality (3.6). �

Proposition 3.3. We have

ψm : ta1... amb1... bm

(u) 7→(t 1...m1...m(u+m)

)−1∙ t 1...m,m+a1...m+am1...m,m+b1...m+bm

(u+m),

where ai, bi ∈ {1, . . . , n}.

Proof. The images of the quantum minors under the automorphism ω are provided

by Theorem 3.2. Therefore, the statement follows from the definition of ψm and the use of

Theorem 3.2 twice. �

In particular, we obtain another description of ψm as an alternative to Lemma 3.1.

Corollary 3.4. For any 1 6 i, j 6 n, we have

ψm : tij(u) 7→(t 1...m1...m(u+m)

)−1∙ t 1...m,m+i1...m,m+j(u+m). (3.9)

�

For any indices i, j = 1, . . . , n, introduce the following series t ]ij(u) with coefficients in

Y(glm+n),

t ]ij(u) = t1...m,m+i1...m,m+j(u) (3.10)

and combine them into the matrix T ](u) = [t ]ij(u)]. Let T(m)(u) be the submatrix of T (u)

determined by the first m rows and columns.

Theorem 3.5 (Quantum Sylvester theorem). The mapping

tij(u) 7→ t ]ij(u), 1 6 i, j 6 n, (3.11)

defines a homomorphism Y(gln)→ Y(glm+n). Moreover, we have the identity

qdetT ](u) = qdet T (u) ∙ qdetT (m)(u− 1) . . . qdetT (m)(u− n+ 1) .

3. QUANTUM SYLVESTER THEOREM FOR Y(glN ) 27

Proof. By Corollary 3.4, we have

ψm : tij(u−m) 7→(qdetT (m)(u)

)−1∙ t ]ij(u). (3.12)

Corollary 2.8 implies that the coefficients of the series qdet T (m)(u) commute with those

of the series t ]ij(v). Hence, since ψm is a homomorphism, we can conclude by the applica-

tion of the shift automorphism (1.20) that the mapping (3.11) defines a homomorphism.

Furthermore, by Proposition 3.3,

ψm : t1... n1... n(u−m) 7→

(qdetT (m)(u)

)−1∙ qdetT (u).

On the other hand, expanding the quantum minor, we obtain from (3.9) that

ψm : t1... n1... n(u−m) 7→

(qdetT (m)(u) . . . qdetT (m)(u− n+ 1)

)−1∙ qdetT ](u),

completing the proof of the desired identity for qdet T ](u). �

We conclude this chapter with a “dual” version of Theorem 3.5. Let us fix an integer

m such that 1 6 m 6 n. For any indices i, j = 1, . . . ,m introduce the following series with

coefficients in Y(gln),

t [ij(u) = ti,m+1... nj,m+1... n(u) (3.13)

and combine them into the matrix T [(u) = [t [ij(u)]. Let T(m+1)(u) be the submatrix of

T (u) obtained by removing the first m rows and columns. The quantum determinant

qdetT(m+1)(u) is well-defined since the assignment

tkl(u) 7→ tk+m,l+m(u) for k, l = 1, . . . , n−m

determines an embedding Y(gln−m)→ Y(gln).

Theorem 3.6. The mapping

tij(u) 7→ t [ij(u), 1 6 i, j 6 m, (3.14)

defines a homomorphism Y(glm)→ Y(gln). moreover, we have the identity

qdetT [(u) = qdet T (u) ∙ qdetT(m+1)(u− 1) . . . qdetT(m+1)(u−m+ 1) .

Proof. This is immediate from Theorem 3.5 by the replacement of m + n by n and

application of the automorphism of Y(gln) given by

υn : tij(u) 7→ tn−i+1,n−j+1(u) (3.15)

which is a particular case of the automorphism (1.21) with the anti-diagonal matrix B =

[δi,n−j+1]. �

Remark 3.7. It is easy to prove that the homomorphisms of Theorems 3.5 and 3.6 are

injective. However, this fact will not be used in the sequel, so we omit the proof. �

CHAPTER 4

Centraliser construction for U(gln)

A slight modification of the proof of Theorem 7.4 yields an alternative method to prove

that the Yangian Y(glm) is embedded into the Olshanski algebra Am; cf. [Mo], [O2].

For any positive integer n consider the general linear Lie algebra gln. The Lie algebra

gln−1 can be identified with the subalgebra of gln spanned by the basis elements Eij with

1 6 i, j 6 n − 1. Fix a nonnegative integer m such that m 6 n and denote by gln,m the

subalgebra in gln spanned by the basis elements Eij with m+1 6 i, j 6 n. The subalgebra

gln,m is isomorphic to gln−m. Let Am(n) denote the centralizer of gln,m in the universal

enveloping algebra U(gln). Let A(n)0 denote the centralizer of the element Enn in U(gln),

let I(n) be the left ideal in U(gln) generated by the elements Ein, i = 1, . . . , n and let J(n)

be the right ideal in U(gln) generated by the elements Eni, i = 1, . . . , n.

Proposition 4.1. I(n)0 := I(n) ∩ A(n)0 is a two-sided ideal in A(n)0 and one has avector space decomposition

A(n)0 = I(n)0 ⊕ U(gln−1).

Proof. By the Poincare–Birkhoff–Witt theorem for the Lie algebra gln, any element

a ∈ U(gln) can be uniquely written as the sum of terms of the form

Ep1n1 ∙ ∙ ∙E

pn−1n,n−1 xE

q11n ∙ ∙ ∙E

qn−1n−1,nE

rnn, (4.1)

where x ∈ U(gln−1). By the commutation relations for U(gln), each term (4.1) is aneigenvector for adEnn with eigenvalue

(p1 + ∙ ∙ ∙+ pn−1)− (q1 + ∙ ∙ ∙+ qn−1). (4.2)

Then a ∈ A(n)0 if and only if for every term (4.1) appearing in a the correspondingexpression (4.2) is zero. Such a term with

∑pi =

∑qi is contained in I(n) iff it is

contained in J(n) iff∑pi 6= 0 or r 6= 0. This shows I(n) ∩ A(n)0 = J(n) ∩ A(n)0 is a

two-sided ideal. Finally, there is at most one nonzero term for which both r and∑pi

vanish: such a term is an element x ∈ U(gln−1). �

Therefore, the projection of A(n)0 onto U(gln−1) with the kernel I(n)0 is an algebra

homomorphism.

Proposition 4.2. If m < n then its restriction to the subalgebra Am(n) defines a

homomorphism

on : Am(n)→ Am(n− 1). (4.3)

29

30 4. CENTRALISER CONSTRUCTION FOR U(glN )

Proof. We just need to show that the image of on|Am(n) lies in Am(n − 1), and thisfollows from the fact that the decomposition of an element a ∈ A(n)0 into sums of theform (4.1) is stable under the action of gln,m. �

Note that the algebra Am(n) inherits the natural filtration of U(gln) and the homomor-

phism on is filtration preserving.

Definition 4.3. The algebra Am is defined as the projective limit of the sequence of

the algebras Am(n), n > m, with respect to the homomorphisms

Am(m)om+1←−−− Am(m+ 1)

om+2←−−− ∙ ∙ ∙on←−−− Am(n)

on+1←−−− ∙ ∙ ∙ ,

where the limit is taken in the category of filtered associative algebras. �

We will use the filtration on Am in the proof of Theorem 4.9. In other words, an element

of the algebra Am is a sequence of the form

a = (am, am+1, . . . , an, . . . )

with an ∈ Am(n), on(an) = an−1 for n > m, and

deg a := supn>mdeg an <∞, (4.4)

where deg an denotes the degree of an in the universal enveloping algebra U(gln). If b =

(bm, bm+1, . . . , bn, . . . ) is another element of Am then the product ab is the sequence

ab = (ambm, am+1bm+1, . . . , anbn, . . . ).

We define the Lie algebra gl∞ as the inductive limit of the Lie algebras gln with respect

to the natural embeddings gln ↪→ gln+1,

gl∞ =⋃

n>1

gln.

Equivalently, gl∞ is the Lie algebra of all complex matrices B = [bij ], where i and j run

over the set of positive integers such that the number of nonzero entries bij is finite. So, gl∞has a basis {Eij}, where i, j > 1 and the Eij satisfy the standard commutation relations(1.3).

By definition, the algebra A0(n) coincides with the center Z(gln) of the universal en-

veloping algebra U(gln). The elements of the algebra A0 can therefore be regarded as

virtual Casimir elements (or virtual Laplace operators) for the Lie algebra gl∞.

Observe that the homomorphisms on are compatible with the natural embeddings

Am(n) ↪→ Am+1(n), that is, the following diagram commutes:

Am(n) −−−→ Am+1(n)

on

y

yon

Am(n− 1) −−−→ Am+1(n− 1).

4. CENTRALISER CONSTRUCTION FOR U(glN ) 31

Therefore we can define an embedding Am ↪→ Am+1 by

(am, am+1, am+2, . . . ) 7→ (am+1, am+2, . . . ).

Definition 4.4. TheOlshanski algebra A = A(gl∞) associated with gl∞ is the inductive

limit of the filtered algebras Am,

A =⋃

m>0

Am. �

Note that the universal enveloping algebra U(gl∞) is canonically embedded into the

Olshanski algebra A. The image of an element x ∈ U(gl∞) in A is the sequence (x, x, . . . ).This is a well-defined map because x belongs to U(glm) for some m and hence to Am(n)

for all n > m.

Proposition 4.5. The center of the Olshanski algebra A coincides with A0.

Proof. It is clear that A0 is contained in the center. Conversely, suppose that a =

(am, am+1, . . . ) ∈ A is a central element. Then it commutes with gl∞ ⊂ A and so an liesin the center of U(gln) for all n > m which implies that a ∈ A0. �

Remark 4.6. It is well-known that the center of the universal enveloping algebra

U(gl∞) is trivial, so that gl∞ has no “genuine” Casimir elements. In contrast to U(gl∞),

the Olshanski algebra A has a large center A0 consisting of the virtual Casimir elements.

As we shall see in the next chapter, these elements act as scalars in the natural class of

highest weight gl∞-modules. From this perspective, the algebra A appears to be a more

natural analogue of the enveloping algebras U(gln) than U(gl∞).

Let us apply the evaluation homomorphism

Y(gln)→ U(gln), T (u) 7→ 1 + Eu−1, (4.5)

see (1.17), to the quantum determinant qdet T (u). We get the polynomial in u−1,

qdet (1 + Eu−1) =∑

σ∈Sn

sgn (σ)(δσ(1)1 + Eσ(1)1u−1)

× (δσ(2)2 + Eσ(2)2 (u− 1)−1) ∙ ∙ ∙ (δσ(n)n + Eσ(n)n (u− n+ 1)

−1), (4.6)

whose coefficients are central in U(gln). Applying the Harish-Chandra homomorphism

(1.6) to the coefficients (see Chapter 1), we find

χ : qdet (1 + Eu−1) 7→ (1 + λ1u−1) ∙ ∙ ∙ (1 + λn(u− n+ 1)

−1).

Hence, the coefficients of the polynomial (4.6) generate the center Z of U(gln).

Some families of generators of the algebra A0 can be constructed by taking an appro-

priate limit as n→∞. Here we shall denote by E the infinite matrix [Eij ] whose rows and


columns are numbered by positive integers while E(n) will denote its submatrix determined

by the first n rows and columns. The series

qdet (1 + E(n)u−1) =∑

p∈Sn

sgn p ∙ (1 + Eu−1)p(1),1 . . . (1 + E(u− n+ 1)−1)p(n),n

in the image of the quantum determinant qdet T (u) under the evaluation homomorphism

(1.17). Write

qdet (1 + E(n)u−1) = 1 + E (n)1 u−1 + E (n)2 u−2 + . . . , E (n)k ∈ U(gln).

We see that all coefficients E (n)k belong to Z(gln). Moreover, by the definition of the

homomorphism on we have

on : qdet (1 + E(n)u−1) 7→ qdet (1 + E(n−1)u−1).

Hence, for any k > 1 we may define a virtual Casimir element Ek ∈ A0 as the sequence

Ek = (E(0)k , E (1)k , E (2)k , . . . ).

In order to get an alternative expression for the virtual Casimir elements Ek, denote by S∞the group of finite permutations of the set of positive integers, so that for any p ∈ S∞ wehave p(l) = l for sufficiently large l. Define the virtual quantum determinant as the formal

power series

qdet (1 + Eu−1) = 1 + E1u−1 + E2u

−2 + . . . .

Using the expression

qdet (1 + Eu−1) =∑

p∈S∞

sgn p ∙ (1 + Eu−1)p(1),1 (1 + E(u− 1)−1)p(2),2 . . . ,

we can regard the coefficients Ek as certain formal series of elements of U(gl∞). For instance,

E1 =∞∑

i=1

Eii,

E2 =∞∑

i=1

(i− 1)Eii +∑

16i<j

(EiiEjj − EjiEij

).

Proposition 4.7. The elements E1, E2, . . . are algebraically independent and generatethe algebra A0. Their images under the isomorphism χ are found by

χ : qdet (1 + Eu−1) 7→∞∏

l=1

u+ λl − lu− l

.


Proof. The second statement is immediate from the definition of the virtual quantum

determinant. This implies that χ(Ek) is the shifted symmetric function found by

1 +∞∑

k=1

χ(Ek)u−k =

∞∏

l=1

u+ λl − lu− l

.

It is clear that χ(Ek) has degree k as an element of Λ∗ and the homogeneous component ofχ(Ek) of degree k coincides with the k-th elementary symmetric function in the variablesλi. Since the elementary symmetric functions are algebraically independent generators of

the algebra Λ, the first statement follows. �

We are now at a stage where we can establish a relation between the Olshanski algebra

Am and the Yangian Y(glm). Now we use the quantum Sylvester theorem (Theorem 3.5).

Taking the composition of the homomorphism (3.14) and the evaluation homomorphism

(4.5) we obtain a homomorphism Y(glm)→ U(gln) which can be written

tij(u) 7→ qdet (1 + Eu−1)BiBj , (4.7)

where we use the quantum determinant of the submatrix of 1 + Eu−1 corresponding to

the rows Bi and columns Bj, where Bi denotes the set {i,m+1, . . . , n}. The image of thishomomorphism lies in the centraliser Am(n), so that we have a homomorphism

ϕn : Y(glm)→ Am(n).

Using the explicit formula for qdet (1 + Eu−1)BiBj as in (4.6), we find that the diagram

Y(glm) Y(glm) ∙ ∙ ∙ Y(glm) ∙ ∙ ∙

ϕm

y ϕm+1

y ϕn

y

Am(m) ←−−−om+1

Am(m+ 1) ←−−− ∙ ∙ ∙ ←−−−on

Am(n) ←−−−on+1

∙ ∙ ∙

is commutative. Note that the image of the generator t(r)ij of Y(glm) under ϕn has degree

6 r for any n. Hence the sequence of homomorphisms (ϕn | n > m) defines an algebra

homomorphism ϕ : Y(glm) → Am which can be written in terms of the virtual quantumdeterminants by

ϕ : tij(u) 7→ qdet (1 + Eu−1)BiBj ,

where Bi now denotes the infinite set {i,m+ 1,m+ 2, . . . }.We will need the following weak form of the Poincare–Birkhoff–Witt theorem for the

Yangian.

Lemma 4.8. The ordered monomials in the generators t(r)ij span Y(glm).

Proof. Use the filtration on the Yangian defined by deg t(r)ij = r. It is immediate from

the defining relations (1.9) that the corresponding graded algebra grY(glm) is commutative.

�


Theorem 4.9. The map ϕ = (ϕm, ϕm+1, . . . ) is an algebra embedding Y(glm) ↪→ Am.

Proof. By Lemma 4.8, the ordered monomials in the generators t(r)ij span Y(glm). To

show that ϕ is an algebra embedding, it will be sufficient to show that the images of these

monomials under ϕ are linearly independent.

Suppose now that the image under ϕ of some nontrivial linear combination of the

ordered monomials in the generators t(r)ij is zero.

The enveloping algebra U(gln) has the usual filtration defined by degEij = 1. Denote

the image of Eij in the corresponding graded algebra grU(gln)∼= S(gln) by xij . We will

identify grU(gln) with the algebra of polynomials in the n2 variables xij , which we combine

into the matrix X = [xij ].

Observe that the image of

ϕn(tij(u)) = qdet T (u)BiBj

under the homomorphism U(gln)→ grU(gln) is the usual determinant

det(1 +Xu−1)BiBj

and write

det(1 +Xu−1)BiBj = δij + λ(1)ij u

−1 + ∙ ∙ ∙+ λ(n−m+1)ij u−n+m−1,

so that λ(r)ij is the sum of all r × r principal minors of XBiBj .

The proof will be completed if we show that given any positive integer K, there exists a

value of n sufficiently great such that the polynomials λij(r) with 1 6 r 6 K and 1 6 i, j 6m are algebraically independent. This will lead to a contradiction with the assumption

that the images of ordered monomials in the generators t(r)ij are linearly dependent.

We now use a standard argument to rephrase the algebraic independence of polynomials

in terms of surjectivity of the related functions. Suppose the polynomials λ(k)ij (X) are

algebraically dependent, so there exists a polynomial F 6= 0 such that F (λ(k)ij ) = 0. Ifwe now consider evaluating the variables xij at n

2 complex values and taking the first

K 6 n−m+ 1 coefficients of det(1 + (X|M)u−1)BiBj we obtain a function

Λn : Cn2 −→ CKm2 (4.8)

given by

(xij | 1 6 i, j 6 n) 7−→ (λ(r)kl | 1 6 k, l 6 m, 1 6 r 6 K).

Evaluating F ◦ Λn at any matrix of complex values M ∈ Cn2 gives zero, that is the image

of Λn is contained in the zero-set V (F ) of F . But V (F ) must be a proper subset of CKm2

because F is nonzero, so that Λn cannot be surjective.

So we see that surjectivity of Λn : Cn2 → CKm2 implies algebraic independence of the

λ(k)ij . We will show that Λn is a surjective function for any n > (K + 1)m.


We first need two auxiliary lemmas. For any r > 1 we let er(z1, . . . , zl) denote the r-thelementary symmetric function in variables z1, . . . , zl with l > r so that

er(z1, . . . , zl) =∑

16i1<∙∙∙<ir6l

zi1 . . . zir .

We also set e0(z1, . . . , zl) = 1 and er(z1, . . . , zl) = 0 for r < 0. Let l be a positive integer

and let α1, . . . , αl be distinct complex parameters. For each i = 1, . . . , l set

eri = er(α1, . . . , αi, . . . , αl),

where the hat indicates the symbol to be omitted. For any 1 6 p 6 l consider the matrix

E =

e01 ∙ ∙ ∙ e0le11 ∙ ∙ ∙ e1l...

. . ....

ep−1,1 ∙ ∙ ∙ ep−1,l

Lemma 4.10. All p× p minors of the matrix E are nonzero.

Proof. Introduce the polynomials Q1(t), . . . , Ql(t) by

Qi(t) = e0i tl−1 + e1i t

l−2 + ∙ ∙ ∙+ el−1,i =l∏

k=1, k 6=i

(t+ αk).

They are linearly independent for if c1Q1(t) + ∙ ∙ ∙+ clQl(t) = 0 then taking t = −αi givesci = 0 since the parameters αi are all distinct.

Suppose now that columns i1, . . . , ip with i1 < ∙ ∙ ∙ < ip of the matrix E are linearlydependent. This implies that a certain nontrivial linear combination Q(t) = d1Qi1(t) +

∙ ∙ ∙+dpQip(t) is a polynomial in t of degree not exceeding l−p−1. However, Q(t) has l−pdistinct roots t = −αj with j ∈ {1, . . . , l} \ {i1, . . . , ip}, and so Q(t) = 0. This contradictsthe linear independence of the polynomials Qi(t). �

Suppose now that E0, . . . , Em−1 are given non-singular p × p matrices with complex

entries.

Lemma 4.11. For any distinct complex numbers β1, . . . , βm the block matrix

E0 β1E1 ∙ ∙ ∙ βm−11 Em−1E0 β2E1 ∙ ∙ ∙ βm−12 Em−1...

.... . .

...

E0 βmE1 ∙ ∙ ∙ βm−1m Em−1

(4.9)

is non-singular.


Proof. Suppose that a linear combination of the rows of the matrix with coefficients

c1, c2, . . . , cpm is zero. Since the rows of each of the matrices Ej are linearly independent,for any index 1 6 i 6 p we have the linear relations

β k1 ci + βk2 cp+i + ∙ ∙ ∙+ β

kmc(m−1)p+i = 0, k = 0, 1, . . . ,m− 1.

As the βj are distinct, for each i the system has only the trivial solution ci = cp+i = ∙ ∙ ∙ =c(m−1)p+i = 0. �

Now, we shall show that the map (4.8) is surjective even when the variables xkl with

m+ 1 6 k 6 n are specialized in the following way:

xkl = βkl, for l = 1, . . . ,m

and

xkl = δklαk, for l = m+ 1, . . . , n

where αm+1, . . . , αn are distinct complex numbers and the βkl are certain complex num-

bers to be chosen below. Under this specialization, for each fixed value of the index

i ∈ {1, . . . ,m} consider the following restriction of the map (4.8)

Λ(i)n : Cn → Cpm+i−1, (4.10)

given by

(xi1, xi2, . . . , xin) 7→

(λ(0)i1 , . . . , λ

(p)i1 , . . . , λ

(0)i,i−1, . . . , λ

(p)i,i−1, λ

(1)ii , . . . , λ

(p)ii , . . . , λ

(1)im, . . . , λ

(p)im),

so that the functions λ(r)ij are defined by the expansions

∣∣∣∣∣∣∣∣∣∣∣

δijt+ xij xi,m+1 xi,m+2 ∙ ∙ ∙ xinβm+1,j t+ αm+1 0 ∙ ∙ ∙ 0

βm+2,j 0 t+ αm+2 ∙ ∙ ∙ 0...

......

. . ....

βn,j 0 0 ∙ ∙ ∙ t+ αn

∣∣∣∣∣∣∣∣∣∣∣

= δijtn−m+1 + tn−mλ

(1)ij + ∙ ∙ ∙+ λ

(n−m+1)ij .

=N+1∑

r=0

tN+1−r((xij − tδij)er(αm+1, . . . , αn)−

m+N∑

k=m+1

βkj xik er−2(αm+1, . . . , αk, . . . , αn)

+ δijer(αm+1, . . . , αk, . . . , αn))(4.11)

More explicitly, we have

λ(r)ij = xij er(αm+1, . . . , αn)−

n∑

k=m+1

βkjxik er−1(αm+1, . . . , αk, . . . , αn)


for j 6= i and

λ(r)ii = xiier−1(αm+1, . . . , αn)

−n∑

k=m+1

βkixik er−1(αm+1, . . . , αk, . . . , αn) + er(αm+1, . . . , αn).

Since the λ(r)ij are affine functions in the variables xi1, . . . , xin, in order to establish

the surjectivity of the map (4.10), it will be sufficient to demonstrate that the rank of

the corresponding coefficient matrix of the linear part is maximal, that is, equal to K.

Writing down the matrix in an explicit form and using the observation that λ(0)ij = xij for

j = 1, . . . , i− 1 we conclude that the claim will follow by proving that the matrix

−βm+1,1e0,m+1 ∙ ∙ ∙ −βn1e0,n−βm+1,1e1,m+1 ∙ ∙ ∙ −βn1e1,n

... ∙ ∙ ∙...

−βm+1,1ep−1,m+1 ∙ ∙ ∙ −βn1ep−1,n... ∙ ∙ ∙

...... ∙ ∙ ∙

...

−βm+1,me0,m+1 ∙ ∙ ∙ −βnme0,n−βm+1,me1,m+1 ∙ ∙ ∙ −βnme1,n

... ∙ ∙ ∙...

−βm+1,mep−1,m+1 ∙ ∙ ∙ −βnmep−1,n

(4.12)

has rank pm, where erk = er(αm+1, . . . , αk, . . . , αn). Now, we use the assumption n−m >Km and Lemma 4.10. Giving appropriate values to the parameters βkl, we may choose a

submatrix of (4.12) of size Km×Km of the form (4.9), where all the K ×K blocks Ei arenon-singular and the parameters βi are distinct. Thus, the claim follows by the application

of Lemma 4.11.

Finally, by allowing the index i in (4.10) to vary and by choosing the parameters βkl in

the same way for each value of i ∈ {1, . . . ,m}, we get a surjective map

Cmn → CKm2 ,

given by

(xij | 1 6 i 6 m, 1 6 j 6 n) 7→ (λ(r)il | 1 6 i, l 6 m, 1 6 r 6 K)

with i > l for r = 0, thus completing the proof. �

As a corollary we have a new proof of the Poincare–Birkhoff–Witt theorem for the

Yangian Y(glm); cf. [BK1], [MNO], [N].


Corollary 4.12. Ordered monomials in the generators t(r)ij form a basis of the Yangian

Y(glm).

Proof. We have shown that the ordered monomials span Y(glm), so it remains to

show they are linearly independent. But linear independence of ordered monomials in the

generators is just the same as algebraic independence of the generators. �

Remark 4.13. The method used for the proof of Theorem 4.9 can be extended to show

that Am∼= A0 ⊗ Y(glm), by demonstrating that the coefficients of the virtual quantum

minors qdet (1+Eu−1)BB together with those of qdet (1+Eu−1)BiBjform an algebraically

independent set in Am, where B denotes {m+ 1, m+ 2, . . . } and Bi denotes {i} ∪ B.

CHAPTER 5

Quantum determinant for Uq(gln)

Now we recall the well-known construction of the quantum determinants for the algebra

Uq(gln); see e.g. [C], [J1], [RTF].

5.1. Fundamental relation

Consider the multiple tensor product Uq(gln) ⊗ (EndCn)⊗r and use the notation of

(1.38). Put

R(u1, . . . , ur) =∏

i<j

Rij(ui, uj), (5.1)

for r > 2, with the product taken in the lexicographical order on the pairs (i, j), i.e.

R(u1, . . . , um) := (Rm−1,m)(Rm−2,mRm−2,m−1) ∙ ∙ ∙ (R1m ∙ ∙ ∙R12), (5.2)

where we abbreviate Rij(ui, uj) by Rij. Then we have the following corollary of (1.40) and

(1.38):

Proposition 5.1 (Fundamental identity).

R(u1, . . . , ur)T1(u1) ∙ ∙ ∙Tr(ur) = Tr(ur) ∙ ∙ ∙T1(u1)R(u1, . . . , ur) (5.3)

Proof. The proof is identical to the proof of (2.4). �

5.2. q-Permutation operator

Define the q-permutation operator P q ∈ End(Cn ⊗ Cn) by

P q =∑

i

Eii ⊗ Eii + q∑

i>j

Eij ⊗ Eji + q−1∑

i<j

Eij ⊗ Eji. (5.4)

Note that (P q)2 = 1. For q = 1 the operator P q turns into the usual permutation operator

P =∑

i,j

Eij ⊗ Eji.

Let si denote the transposition (i, i+ 1), for i = 1, . . . , r − 1.Define P q

si:= P q

i,i+1 i.e. Pq operating in the ith and (i+1)th copies of Cn. If σ = si1 ∙ ∙ ∙ sil

is a reduced decomposition of an element σ ∈ Sr we set Pqσ = P

qsi1∙ ∙ ∙P q

sil.

Notation. Denote by e1, . . . , en the canonical basis vectors of Cn. To simplify formulae

we will use the notation |a1, . . . , ar〉 := ea1 ⊗ ∙ ∙ ∙ ⊗ ear ∈ (Cn)⊗r.

39

40 5. QUANTUM DETERMINANT FOR Uq(glN )

Lemma 5.2. For any indices a1 < ∙ ∙ ∙ < ar we have

P qσ |a1, . . . , ar〉 = q

`(σ)∣∣aσ−1(1), . . . , aσ−1(r)

⟩, (5.5)

where `(σ) denotes the length of the permutation σ.

Proof. Proof by induction on `(σ). If σ = 1 we are done. Else if σ = si1τ =

si1si2 ∙ ∙ ∙ si`(σ) is a reduced expression for σ, then

P qσ (|a1, . . . , ar〉) = P

qsi1P qτ (|a1, . . . , ar〉) = q

`(σ)−1 P qsi1(∣∣aτ−1(1), . . . , aτ−1(r)

⟩)

Now σ−1 = τ−1si1 is also reduced, and by a property of reduced expressions (see for

example [H] p. 50), σ−1(i1) > σ−1(i1 + 1), so that τ−1(i1) < τ−1(i1 + 1). Then by (5.4),

P qs1(∣∣aτ−1(1), . . . , aτ−1(r)

⟩) = q

∣∣aσ−1(1), . . . , aσ−1(r)

⟩.

�

This implies a more general formula: for any τ ∈ Sr we have

P qσ (∣∣aτ(1), . . . , aτ(r)

⟩) = q−`(τ)P q

σ Pqτ−1(|a1, . . . , ar〉)

= q`(στ−1)−`(τ)

∣∣aτσ−1(1), . . . , aτσ−1(r)

⟩.

(5.6)

If ω = ω−1 : i 7→ n − i + 1 denotes the longest element of Sn, then we have `(σω) =

`(ω)− `(σ) for all σ ∈ Sn. Substituting ω for τ in the above gives the corollary

P qσ (|an, . . . , a1〉) = q

−`(σ)∣∣aσ(n), . . . , aσ(1)

⟩. (5.7)

It is easily seen that (P qi )2 = 1, and that P q

i and Pqj commute for |i− j| > 1. In fact P

q

defines a representation of the symmetric group on (Cn)⊗r; it remains to demonstrate that

the P qi satisfy the braid relations

P qi P

qi+1P

qi = P

qi+1P

qi P

qi+1, 1 6 i < n. (5.8)

An explicit calculation is straight-forward, albeit tedious; it is more illuminating to make

the following observation. Set

ei1∙∙∙ ir := q#inv{i1∙∙∙ ir}ei1 ⊗ ei2 ⊗ ∙ ∙ ∙ ⊗ eir , 1 6 i1, . . . , ir 6 n, (5.9)

where

#inv{i1 ∙ ∙ ∙ ir} = #{α < β | iα > iβ} (5.10)

is the number of strict inversions in the sequence i1, . . . , ir. Then Pqσ acts on this new basis

simply as

P qσ ∙ ei1... ir = eiσ−1(1)∙∙∙ iσ−1(r) .

To see this it is enough to check for the generators of Sr, i.e. to check that Pqα,α+1 ∙ei1∙∙∙ ir =

ei1∙∙∙ iα+1iα∙∙∙ ir for α = 1, . . . , r − 1. But

P qα,α+1 ∙ ei1∙∙∙ ir = q

#inv{i1∙∙∙ir}+Cei1 ⊗ ∙ ∙ ∙ ⊗ eiα+1 ⊗ eiα ⊗ ∙ ∙ ∙ ⊗ eik ,

5.3. Q-ANTISYMMETRIZER AND QUANTUM DETERMINANT 41

where C = 1, 0 or −1 according as to whether iα < iα+1, iα = iα+1 or iα > iα+1. Clearly,

#inv{i1, ∙ ∙ ∙ , iα+1, iα, ∙ ∙ ∙ ir} = #inv{i1, ∙ ∙ ∙ , ir}+ C

so we are done.

In other words we have proved

Lemma 5.3. The map σ 7→ P qσ is a representation of the symmetric group on (C

n)⊗r

which is isomorphic to the standard representation. The change of basis map

ei1 ⊗ ∙ ∙ ∙ ⊗ eik 7−→ ei1∙∙∙ ik

provides an intertwining operator between these representations.

Remark 5.4. This provides another proof of (5.2), (5.6) and (5.7).

5.3. q-Antisymmetrizer and quantum determinant

We denote by Aqr the q-antisymmetrizer

Aqr =∑

σ∈Sr

sgn σ ∙ P qσ , (5.11)

where sgn σ = (−1)`(σ) is the sign of σ. That is, Aqr is the image in (EndCn)r, under the

representation σ 7→ P qσ of Sr, of the antisymmetrizer

∑σ∈Sr

sgn σ ∙σ in the group algebra.The above formulae imply that for any τ ∈ Sr we have

Aqr(∣∣aτ(1), . . . , aτ(r)

⟩) = (−q)−l(τ)Aqr(|a1, . . . , ar〉).

Lemma 5.5. We have the following relation in End(Cn)⊗r:=

R(1, q−2, . . . , q−2r+2) =∏

06i<j6r−1

(q−2i − q−2j)Aqr. (5.12)

Proof. We use induction on r. Using (1.39) we have

R(1, q−2) = (1− q−2)(1− P q),

which proves the case r = 2. Then, writing Rij for Rij(q2j−2i) we have

R(1, . . . , q−2r+2) = R(1, . . . , q2r−4)R1r ∙ ∙ ∙Rr−1,r

=∏

06i<j6r−2

(q−2i − q−2j)Aqr−1R1r ∙ ∙ ∙Rr−1,r, by induction

=1

(r − 2)!

∏

06i<j6r−2

(q−2i − q−2j)Aqr−1R1rAq{2,...,r−1}R2r ∙ ∙ ∙Rr−1,r,

(5.13)

where we introduce the antisymmetrizer on the subset of indices {2, . . . , r}, and observethat

AqaAqb = A

qbA

qa =1

a!Aqb for a 6 b.


Now, using induction three times in succession, we have

Aq{2,...,r−1}R2r ∙ ∙ ∙Rr−1,r =∏

16i<j6r−2

(q−2i − q−2j)−1R(q−2, . . . , q−2r−4)R2r ∙ ∙ ∙Rr−1,r

=∏

16i<j6r−2

(q−2i − q−2j)−1R(q−2, . . . , q−2r+2)

=∏

16i<j6r−2

(q−2i − q−2j)−1∏

16i<j6r−1

(q−2i − q−2j)Aq{2,...,r}

=r−2∏

i=1

(q−2i − q−2(r−1))Aq{2,...,r}.

(5.14)

Now observe that Aqk = (1− Pq1k − ∙ ∙ ∙ − P

qk−1,k)A

qk−1. So

R(u1, . . . , ur) =((r − 2)!

)−1(1− q−2r+2

)−1∏

06i<j6r−1

(q−2i − q−2j)Aqr−1R1rAq{2,...,r}

=((r − 2)!

)−1(1− q−2r+2

)−1∏

06i<j6r−1

(q−2i − q−2j)Aqr−1R1rAq{2,...,r−1}

× (1− P q(2r) − . . . P

q(r−1,r))

=(1− q−2r+2

)−1∏

06i<j6r−1

(q−2i − q−2j)Aqr−1R1r(1− Pq(2r) − . . . P

q(r−1,r))

(5.15)

Since

R(u1, . . . , ur) =1

(r − 1)!R(u1, . . . , ur)A

qr−1 =

1

(r − 1)!R(u1, . . . , ur)A

q{2,...,r}, (5.16)

we see that it is sufficient to apply R(u1, . . . , ur) to the vector e1 ⊗ ∙ ∙ ∙ ⊗ er.

Aqr−1R1r(1− Pq(2r) − . . . P

q(r−1,r))e1 ⊗ ∙ ∙ ∙ ⊗ er

= Aqr−1[(1− q−2r+2)

∑

i 6=j

(Eii)1(Ejj)r + (q−1 − q)

∑

i>j

(Eij)1(Eji)r]

× (e1 ⊗ ∙ ∙ ∙ ⊗ er −r−2∑

i=1

q2i−1e1 ⊗ ∙ ∙ ∙ er−i ∙ ∙ ∙ ⊗ er−1 ⊗ er−i)

(5.17)

= Aqr−1[(1− q−2r+2)(e1 ⊗ ∙ ∙ ∙ ⊗ er −

r−2∑

i=1

q2i−1e1 ⊗ ∙ ∙ ∙ er−i ∙ ∙ ∙ ⊗ er ⊗ er−i)

+ (q−1 − q)er ⊗ e2 ⊗ ∙ ∙ ∙ ⊗ er−1 ⊗ e1

− (q−1 − q)r−2∑

i=1

q2i−1er−i ⊗ e2 ⊗ ∙ ∙ ∙ er−i ∙ ∙ ∙ ⊗ er ⊗ e1].

(5.18)


Continuing with straightforward calculations we arrive at∑

σ∈Sr

(−q)l(σ)eσ(1) ⊗ ∙ ∙ ∙ ⊗ eσ(r) = Ar(e1 ⊗ ∙ ∙ ∙ ⊗ er). �

Now (5.3) implies

Aqr T1(u) ∙ ∙ ∙Tr(q−2r+2u) = Tr(q

−2r+2u) ∙ ∙ ∙T1(u)Aqr (5.19)

Take r = n in this equation. The operator Aqn on (Cn)⊗n has a one-dimensional image,

so the element (5.19) with r = n equals Aqn times a scalar series with coefficients in Yq(gln).

This prompts the following definition.

Definition 5.6. The quantum determinant of the matrix T (u) with coefficients in

Yq(gln) is the formal series

qdetT (u) = d(u) = d0 + d1u−1 + d2u

−2 + . . .

such that the element (5.19) with r = n, equals An qdetT (u).

Lemma 5.7. We have the following explicit formulae for the quantum determinant .

qdetT (u) =∑

σ

(−q)−`(σ) tσ(1)1(u) ∙ ∙ ∙ tσ(n)n(q−2n+2u)

=∑

σ

(−q)−`(σ) t1σ(1)(q−2n+2u) ∙ ∙ ∙ tnσ(n)(u)

=∑

σ

(−q)+`(σ) tσ(n)n(u) ∙ ∙ ∙ tσ(1)1(q−2n+2u)

=∑

σ

(−q)+`(σ) tnσ(n)(q−2n+2u) ∙ ∙ ∙ t1σ(1)(u).

(5.20)

Proof. Applying the left hand side of 5.19 to |b1, . . . , br〉, we have

Aqr T1(u) ∙ ∙ ∙Tr(q−2r+2u) |b1, . . . , br〉

=n∑

i1,...,ir=1

ti1b1(u) ∙ ∙ ∙ tirbr(q−2r+2u)Aqr |i1, . . . , ir〉

=∑

σ∈Sr

tσ(1)b1(u) ∙ ∙ ∙ tσ(r)br(q−2r+2u)Aqr |σ(1), . . . , σ(r)〉

=∑

σ

q−`(σ)tσ(1)b1(u) ∙ ∙ ∙ tσ(r)br(q−2r+2u)AqrP

qσ−1|1, . . . , r〉

=∑

σ

(−q)−`(σ)tσ(1)b1(u) ∙ ∙ ∙ tσ(r)br(q−2r+2u)Aqr |1, . . . , r〉

which gives the first equation. If we instead apply the left hand side of (5.19) to |n, . . . , 1〉,we get the third line.


Let’s apply d(u)Aqn = Tn(q−2n+2u) ∙ ∙ ∙T1(u)Aqn to |n, . . . , 1〉. On the left hand side we

have

d(u)Aqn |n, . . . , 1〉 = d(u)∑

σ

(−q)−`(σ) |σ(n), . . . , σ(1)〉

while the right gives

Tn(q−2n+2u) ∙ ∙ ∙T1(u)A

qn |n, . . . , 1〉

=∑

~i,~j

tinjn(q−2n+2u) ∙ ∙ ∙ ti1j1(u)Ei1j1 ⊗ ∙ ∙ ∙ ⊗ Einjn

∑

σ

(−q)−`(σ) |σ(n), . . . , σ(1)〉

=∑

σ,~i

(−q)−`(σ)tinσ(1)(q−2n+2u) ∙ ∙ ∙ ti1σ(n)(u) |i1, . . . , in〉

(5.21)

Comparing coefficients at |n, . . . , 1〉 gives the second line. Applying

d(u)Aqn = Tn(q−2n+2u) ∙ ∙ ∙T1(u)A

qn

to |1, . . . , n〉 and comparing coefficients at |1, . . . , n〉 gives the last equation. �

More generally, for m 6 n we can define the m × m quantum minors as the matrix

elements of the operator (5.19). That is, (5.19) can be written asn∑

ai,bi=1

t a1∙∙∙ arb1∙∙∙ br (u)⊗ Ea1b1 ⊗ ∙ ∙ ∙ ⊗ Earbr (5.22)

for some elements t a1∙∙∙ arb1∙∙∙ br (u) of Uq(gln)[[u−1]] which we call the quantum minors.

They are given by the following formulae, which are obvious generalisations of the

formulae in Lemma 5.7. If a1 < ∙ ∙ ∙ < ar then

t a1∙∙∙ arb1∙∙∙ br (u) =∑

σ∈Sr

(−q)−l(σ) ∙ taσ(1)b1(u) ∙ ∙ ∙ taσ(r)br(q−2r+2u). (5.23)

Also, for any τ ∈ Sr we have

taτ(1)∙∙∙ aτ(r)b1∙∙∙ br (u) = (−q)l(τ)t a1∙∙∙ arb1∙∙∙ br (u). (5.24)

If b1 < ∙ ∙ ∙ < br (and the ai are arbitrary) then

t a1∙∙∙ arb1∙∙∙ br (u) =∑

σ∈Sr

(−q)l(σ) ∙ tarbσ(r)(q−2r+2u) ∙ ∙ ∙ ta1bσ(1)(u), (5.25)

For any τ ∈ Sr we have

t a1∙∙∙ arbτ(1)∙∙∙ bτ(r)(u) = (−q)−l(τ)t a1∙∙∙ arb1∙∙∙ br (u). (5.26)

Moreover, the quantum minor is zero if two top or two bottom indices are equal.

As the defining relations for the generators t(r)ij have the same matrix form as for the

t(r)ij , the above argument can be applied to define the respective quantum determinant

qdetT (u) = d0 + d1 u+ d2 u2 + . . .


and quantum minors

ta1∙∙∙ arb1∙∙∙ br (u) ∈ Uq(gln)[[u]].

They are given by the same formulae (5.23) and (5.25), where the tij(u) are respectively

replaced with tij(u). Indeed, we have the relation

Aqr T 1(u) ∙ ∙ ∙T r(q−2r+2u) = T r(q

−2r+2u) ∙ ∙ ∙T 1(u)Aqr (5.27)

analogous to (5.19) so that both sides are equal to∑

ai,bi

ta1∙∙∙ arb1∙∙∙ br (u)⊗ Ea1b1 ⊗ ∙ ∙ ∙ ⊗ Earbr . (5.28)

Clearly the quantum determinants of T (u) and T (u) can be written

qdetT (u) = t 1∙∙∙n1∙∙∙n(u), qdetT (u) = t 1∙∙∙n1∙∙∙n(u). (5.29)

The following proposition is well-known.

Proposition 5.8.

[tcidj(u), tc1∙∙∙ crd1∙∙∙ dr(v)] = 0, [tcidj(u), t

c1∙∙∙ crd1∙∙∙ dr(v)] = 0, (5.30)

and the same holds with tcidj(u) replaced by tcidj(u).

Proof. The proof is analogous to that of Proposition 2.8 �

Due to (5.30), the coefficients dk and dk of the quantum determinants of the matrices

T (u) and T (u) belong to the center of the algebra Uq(gln). Furthermore, we have the

relation d0d0 = d0d0 = 1 which is implied by (1.38).

CHAPTER 6

Quantum Sylvester theorem for Uq(gln)

In this chapter we give a q-analogue of the classical Sylvester theorem, where ordinary

minors are replaced by quantum minors of the matrices T (u) or T (u). Following [KL], for

its proof we employ a certain q-analogue of the complimentary minor identity.

We shall also be using the algebra Uq−1(gln). In order to distinguish its generators from

those of the algebra Uq(gln) we mark them by the symbol◦. In particular, t ◦ij(u) and t

◦ij(u)

will denote the corresponding generating series while T ◦(u) and T◦(u) will stand for the

generator matrices.

Proposition 6.1. The mapping

ωn : T (u) 7→ T ◦(u)−1, T (u) 7→ T◦(u)−1

defines an isomorphism ωn : Uq(gln)→ Uq−1(gln).

Proof. The matrix T ◦(u) satisfies the relation

Rq−1(u, v)T ◦1 (u)T◦2 (v) = T

◦2 (v)T

◦1 (u)R

q−1(u, v).

Multiplying both sides on the left by the product T ◦1 (u)−1T ◦2 (v)

−1 and on the right by

T ◦2 (v)−1T ◦1 (u)

−1, then conjugating by P and swapping the parameters u and v gives

PRq−1(v, u)P T ◦1 (u)−1T ◦2 (v)

−1 = T ◦2 (v)−1T ◦1 (u)

−1PRq−1(v, u)P.

Observing that PRq−1(v, u)P = −Rq(u, v), and applying the same argument to the two

remaining matrix relations in (1.38), we conclude that ωn defines a homomorphism. This

map is obviously invertible with the inverse given by

ω−1n : T◦(u) 7→ T (u)−1, T

◦(u) 7→ T (u)−1

so that ωn is an isomorphism. �

We now prove an analogue of the complementary minor identity (3.4) for Uq(gln)

Theorem 6.2. We have the identities

qdetT (u) ∙ ω−1n(t ◦

jm+1...jnim+1...in

(q−2n+2u))= (−q)l(j)−l(i) ∙ t i1...imj1...jm

(u),

qdetT (u) ∙ ω−1n(t◦ jm+1...jnim+1...in

(q−2n+2u))= (−q)l(j)−l(i) ∙ t i1...imj1...jm

(u),

47

48 6. QUANTUM SYLVESTER THEOREM FOR Uq(glN )

where t ◦jm+1...jnim+1...in

(v) and t ◦jm+1...jnim+1...in

(v) denote the quantum minors in the quantum affine

algebra Uq−1(gln).

Proof. Both identities are verified in the same way so we only consider the first one.

By the definition of the quantum determinant,

qdetT (u)Aqn = Aqn T1 . . . Tn, (6.1)

where Ti = Ti(q−2i+2u) for i = 1, . . . , n. Let us multiply both sides of (6.1) by T−1n . . . T−1m+1

from the right. Then (6.1) takes the form

qdetT (u)Aqn T−1n . . . T−1m+1 = A

qn T1 . . . Tm . (6.2)

Now we apply both sides of (6.2) to the basis vector

ej1 ⊗ ∙ ∙ ∙ ⊗ ejm ⊗ eim+1 ⊗ ∙ ∙ ∙ ⊗ ein ∈ (Cn)⊗n. (6.3)

The right hand side gives

Aqn∑

a1,...,am

ta1j1(u) . . . tamjm(q−2m+2u)ea1 ⊗ ∙ ∙ ∙ ⊗ eam ⊗ eim+1 ⊗ ∙ ∙ ∙ ⊗ ein . (6.4)

The summation here can obviously be restricted to the sequences a1, . . . , am which are

permutations of the sequence i1, . . . , im . Consider the vector Aqn(e1 ⊗ . . . ⊗ en). The sum

(6.4) is proportional to this vector, with the coefficient

(−q)−l(i) ∙ t i1...imj1...jm(u) .

We shall use the notation t ′ij(u) for the entries of the matrix T−1(u). By applying the left

hand side of (6.2) to the basis vector (6.3), we obtain

Aqn∑

bm+1,...,bn

t ′bnin(q−2n+2u) . . . t ′bm+1im+1(q

−2mu)ej1 ⊗ ∙ ∙ ∙ ⊗ ejm ⊗ ebm+1 ⊗ ∙ ∙ ∙ ⊗ ebn .

Here the coefficient of Aqn(e1 ⊗ . . .⊗ en) equals

(−q)−l(j) ∙∑

σ

(−q)−l(σ) t ′jσ(n)in(q−2n+2u) . . . t ′jσ(m+1)im+1(q

−2mu),

summed over permutations σ of the set {m+ 1, . . . , n}. This coefficient can be written as

(−q)−l(j) ∙ ω−1n(t ◦

jm+1...jnim+1...in

(q−2n+2u)).

Thus the desired identity follows from (6.2). �

Corollary 6.3. We have the identities

qdetT (u) ∙ ω−1n(qdetT ◦(q−2n+2u)

)= 1,

qdetT (u) ∙ ω−1n(qdetT

◦(q−2n+2u)

)= 1,

where qdetT ◦(v) and qdetT◦(v) denote the quantum determinants in Uq−1(gln). �

6. QUANTUM SYLVESTER THEOREM FOR Uq(glN ) 49

For any 0 6 m 6 n introduce the homomorphism ım,n

ım,n : Uq−1(glm)→ Uq−1(gln) (6.5)

which takes the coefficients of the series t◦ij(u) and t◦ij(u) to the respective elements of

Uq−1(gln) with the same name. Consider the composition

ψm,n = ω−1n ◦ ım,n ◦ ωm (6.6)

which is an algebra homomorphism

ψm,n : Uq(glm)→ Uq(gln).

The action of ψm,n on the quantum minors is described in the next lemma.

Lemma 6.4. We have

ψm,n : ta1... arb1... br

(u) 7→(tm+1... nm+1... n(q

2n−2mu))−1∙ t a1... ar,m+1... nb1... br,m+1... n

(q2n−2mu),

where ai, bi ∈ {1, . . . ,m}, and the same formula holds with the minors in the tij(u) replacedwith the respective minors in the tij(u).

Proof. Due to (5.24) and (5.26) we may assume that a1 < ∙ ∙ ∙ < ar and b1 < ∙ ∙ ∙ < br.

By Theorem 6.2 and Corollary 6.3 we have

ωm : ta1... arb1... br

(u) 7→ (−q)l(a)−l(b) ∙ t ◦ 1...m1...m(q−2m+2u)−1 ∙ t ◦ br+1...bmar+1...am

(q−2m+2u),

where ar+1 < ∙ ∙ ∙ < am and br+1 < ∙ ∙ ∙ < bm are respective complementary elements to

the sets {a1, . . . , ar} and {b1, . . . , br} in {1, . . . ,m}, while a and b denote the permutations(a1, . . . , am) and (b1, . . . , bm) of the sequence (1, . . . ,m). Now, applying the homomorphism

ım,n we may regard the quantum minors on the right hand side as series with coefficients

in the algebra Uq−1(gln). The proof is completed by another application of Theorem 6.2.

The same argument proves the corresponding formula for the minors in the tij(u). �

In particular, we have the following explicit formulae for the images of the generators

of Uq(gln).

Corollary 6.5. For any 1 6 i, j 6 m, we have

ψm,n : tij(u) 7→(tm+1... nm+1... n(q

2n−2mu))−1∙ t i,m+1... nj,m+1... n(q

2n−2mu),

and the same formula holds with the image of the tij(u) where the minors in the tij(u) are

replaced with the respective minors in the tij(u). �

For any 1 6 i, j 6 m, introduce the following series with coefficients in the q-Yangian

Yq(gln)

t [ij(u) = ti,m+1... nj,m+1... n(u)

and combine them into the matrix T [(u) = [t [ij(u)]. Let T (u)QQ be the submatrix of T (u)

whose rows and columns are numbered by the elements of the set Q = {m+ 1, . . . , n}.

50 6. QUANTUM SYLVESTER THEOREM FOR Uq(glN )

The following is an analogue of the Sylvester theorem for the q-Yangian.


tij(u) 7→ t [ij(u), 1 6 i, j 6 m, (6.7)

defines a homomorphism Yq(glm)→ Yq(gln). Moreover, we have the identity

qdetT [(u) = qdet T (u) ∙ qdetT (q−2u)QQ . . . qdetT (q−2m+2u)QQ.

Proof. By Corollary 6.5, we have

ψm,n : tij(q−2n+2mu) 7→

(qdetT (u)QQ

)−1∙ t [ij(u). (6.8)

Relation (5.30) implies that the coefficients of the series qdet T (u)QQ commute with those of

the series t [ij(v). Hence, since ψm,n is a homomorphism, we can conclude by the application

of the automorphism (1.48) that the mapping (6.7) defines a homomorphism. Furthermore,

by Lemma 6.4,

ψm,n : t1...m1...m(q

−2n+2mu) 7→(qdetT (u)QQ

)−1∙ qdetT (u).

On the other hand, expanding the quantum minor, we obtain from (6.8) that

ψm,n : t1...m1...m(q

−2n+2mu) 7→(qdetT (u)QQ . . . qdetT (q

−2m+2u)QQ)−1∙ qdetT [(u),

completing the proof of the desired identity for qdet T [(u). �

We shall also need a dual version of the quantum Sylvester theorem. Let m > 0 andn > 1. Instead of the homomorphism ım,n defined in (6.5), consider another homomorphism

m : Uq−1(gln)→ Uq−1(glm+n) (6.9)

which takes the coefficients of the series t ◦ij(u) and t◦ij(u) to the respective coefficients of

the series t ◦m+i,m+j(u) and t◦m+i,m+j(u). Consider the composition

φm = ω−1m+n ◦ m ◦ ωn. (6.10)

Then φm is an algebra homomorphism

φm : Uq(gln)→ Uq(glm+n).

The corresponding version of Lemma 6.4 now takes the form

Lemma 6.7. We have

φm : ta1... arb1... br

(u) 7→(t 1...m1...m(q

2mu))−1∙ t 1...m,m+a1...m+ar1...m,m+b1...m+br

(q2mu),

where ai, bi ∈ {1, . . . , n}, and the same formula holds with the minors in the tij(u) replacedwith the respective minors in the tij(u). �

6. QUANTUM SYLVESTER THEOREM FOR Uq(glN ) 51

For any indices 1 6 i, j 6 n introduce the following series with coefficients in the

q-Yangian Yq(glm+n),

t ]ij(u) = t1...m,m+i1...m,m+j(u),

and combine them into the matrix T ](u) = [t ]ij(u)]. Let T (u)PP be the submatrix of T (u)

whose rows and columns are numbered by the elements of the set P = {1, . . . ,m}. Thefollowing theorem is proved in the same way as Theorem 6.6.


tij(u) 7→ t ]ij(u), 1 6 i, j 6 n, (6.11)

defines a homomorphism Yq(gln)→ Yq(glm+n). Moreover, we have the identity

qdetT ](u) = qdet T (u) ∙ qdetT (q−2u)PP . . . qdetT (q−2n+2u)PP . �

Remark 6.9. The obvious analogues of Theorems 6.6 and 6.8 for the subalgebras of

Uq(gln) generated by the elements t(r)ij and tii can be proved by the same argument. �

We also point out a corollary to be used in Chapter 9. It follows from Lemma 6.7 with

the use of the automorphism (1.48).

Corollary 6.10. The mapping

tij(u) 7→(t 1...m1...m(u)

)−1∙ t 1...m,m+i1...m, ,m+j(u),

tij(u) 7→(t1...m1...m(u)

)−1∙ t 1...m,m+i1...m, ,m+j(u),

(6.12)

defines a homomorphism Uq(gln) → Uq(glm+n). Moreover, the images of the quantumminors under the homomorphism (6.12) are found by

t a1... arb1... br(u) 7→

(t 1...m1...m(u)

)−1∙ t 1...m,m+a1...m+ar1...m,m+b1...m+br

(u),

where ai, bi ∈ {1, . . . , n}, and the same formula holds with the minors in the tij(u) replacedwith the respective minors in the tij(u). �

CHAPTER 7

Centraliser construction for Uq(gln)

In this chapter we consider q as a formal variable so that the quantum algebras are

regarded as algebras over C(q). Denote by Uq(gln) the subalgebra of Uq(gln) generated bythe elements

τij = tij tjj , i > j and τij = tij tjj , i 6 j. (7.1)

Since in the algebra Uq(gln) we have

qδja tia tjj = qδij tjj tia and qδja tia tjj = q

δij tjj tia,

we may regard Uq(gln) as an associative algebra generated by the elements τij with i > j

and τij with i 6 j subject to the defining relations

qδij+δja τia τjb − qδib+δab τjb τia = (q − q

−1) qδia(δb<a − δi<j) τja τib,


−1) qδia(δb<a − δi<j) τja τib,


−1) qδia(δb<a τja τib − δi<j τja τib

),

where

τij = τji = 0, 1 6 i < j 6 n, and τii = 1, i = 1, . . . , n.

For any positive integer n the algebra Uq(gln−1) can be identified with a subalgebra of

Uq(gln) generated by the elements τij and τij with 1 6 i, j 6 n− 1.Fix a nonnegative integer m such that m 6 n and denote by Uq(gln,m) the subalgebra

of Uq(gln) generated by the elements τij and τij with m+ 1 6 i, j 6 n. This subalgebra is

isomorphic to Uq(gln−m). Let Am(n) denote the centralizer of Uq(gln,m) in Uq(gln). Also,

let A(n)0 denote the centralizer of the element τnn in Uq(gln) and let I(n) be the left ideal

in Uq(gln) generated by the elements τin, i = 1, . . . , n. Then the Poincare–Birkhoff–Witt

theorem for the algebra Uq(gln) implies that I(n)0 = I(n) ∩ A(n)0 is a two-sided ideal in

A(n)0 and one has a vector space decomposition

A(n)0 = I(n)0 ⊕ Uq(gln−1).

Therefore, the projection of A(n)0 onto Uq(gln−1) with the kernel I(n)0 is an algebra homo-

morphism. If m < n then its restriction to the subalgebra Am(n) defines a homomorphism

on : Am(n)→ Am(n− 1). (7.2)

53

54 7. CENTRALISER CONSTRUCTION FOR Uq(glN )

Note that the algebra Am(n) inherits the filtration of Uq(gln) defined by

deg τij = 0, i > j, and deg τij = 1, i 6 j.

Clearly, the homomorphism on is filtration preserving.

Definition 7.1. The Olshanski algebra Am is defined as the projective limit of the

sequence of the algebras Am(n), n > m, with respect to the homomorphisms

Am(m)om+1←−−− Am(m+ 1)

om+2←−−− . . .on←−−− Am(n)

on+1←−−− . . . ,

where the limit is taken in the category of filtered associative algebras. �

An element of the algebra Am is a sequence of the form a = (am, am+1, . . . , an, . . . ) with

an ∈ Am(n), on(an) = an−1 for n > m, and

deg a = supn>mdeg an <∞, (7.3)

where deg an denotes the degree of an in Uq(gln). If b = (bm, bm+1, . . . , bn, . . . ) is another

element of Am then the product ab is the sequence

ab = (ambm, am+1bm+1, . . . , anbn, . . . ).

We define the algebra Uq(gl∞) as the inductive limit of the algebras Uq(gln) with respect

to the natural embeddings Uq(gln) ↪→ Uq(gln+1),

Uq(gl∞) =⋃

n>1

Uq(gln).

Note that the algebra A0(n) coincides with the center of Uq(gln). The elements of the

algebra A0 can therefore be regarded as virtual Casimir elements for the algebra Uq(gl∞).

Let us apply the evaluation homomorphism

Yq(gln)→ Uq(gln), T (u) 7→ T − T u−1, (7.4)

see (1.45), to the quantum determinant qdet T (u). We get the polynomial in u−1,

qdet (T − T u−1) =∑

σ∈Sn

(−q)−l(σ) (tσ(1)1 − tσ(1)1u−1)

× (tσ(2)2 − tσ(2)2q2u−1) . . . (tσ(n)n − tσ(n)nq

2n−2u−1), (7.5)

whose coefficients are central in Uq(gln). Applying the Harish-Chandra homomorphism

(1.35) to the coefficients (see Chapter 1), we find

χ : qdet (T − T u−1) 7→ qn(n−1)/2 ∙ (x1 − x−11 u

−1) . . . (xn − x−1n u−1).

Hence, the coefficients of the polynomial (7.5) generate the center Zq of Uq(gln).

The product

dn(u) = qdet (T − T u−1) ∙ t11 . . . tnn

7. CENTRALISER CONSTRUCTION FOR Uq(glN ) 55

is a polynomial in u−1 with constant term 1 whose coefficients belong to the center of the

subalgebra Uq(gln). Explicitly, this polynomial can be written as

dn(u) =∑

σ∈Sn

(−q)−l(σ)qind(σ) (τσ(1)1 − τσ(1)1u−1) . . . (τσ(n)n − τσ(n)nq

2n−2u−1), (7.6)

where ind(σ) = ] {i = 1, . . . , n | σ(i) < i}. Write

dn(u) = 1 + d(1)n u−1 + ∙ ∙ ∙+ d (n)n u−n.

Proposition 7.2. The elements d(1)n , . . . , d

(n)n are algebraically independent and gen-

erate the center of the algebra Uq(gln).

Proof. Any central element z of Uq(gln) must commute with τ11, . . . , τnn. Using the

isomorphism (1.34), and writing z as a linear combination of the corresponding basis

monomials, we conclude that z must also commute with t11, . . . , tnn. Therefore, z belongs

to the center Zq of Uq(gln). Hence, z is a polynomial in t0, t−10 , d

(1)n , . . . , d

(n)n , where t0 =

t11 . . . tnn. However, such a polynomial does not belong to the subalgebra Uq(gln) unless it

only contains non-positive even powers of t0. Since t−20 coincides with d

(n)n up to a nonzero

constant factor, this proves that the center of Uq(gln) is generated by d(1)n , . . . , d

(n)n .

Finally, the image of dn(u) under the Harish-Chandra homomorphism is given by

χ : dn(u) 7→ (1− x−21 u−1) . . . (1− x−2n u−1),

thus proving the algebraic independence of d(1)n , . . . , d

(n)n . �

Applying the homomorphism (7.2) with m = 0 to the coefficients of the polynomial

dn(u) we find immediately that

on : dn(u) 7→ dn−1(u).

Since the degree of the element d(k)n does not exceed k for all n > k, we may define a virtual

Casimir element d (k) ∈ A0 for any k > 1 as the sequence

d (k) = (d (k)n | n > 0),

where we set d(k)n = 0 for n < k. In order to get an alternative expression for the virtual

Casimir elements d (k), denote by S∞ the group of finite permutations of the set of positive

integers, so that for any p ∈ S∞ we have p(l) = l for all sufficiently large l. Define the

virtual quantum determinant as the formal power series

d(u) = 1 + d (1)u−1 + d (2)u−2 + . . . .

Using the expression

d(u) =∑

σ∈S∞

(−q)−l(σ)qind(σ) (τσ(1)1 − τσ(1)1u−1)(τσ(2)2 − τσ(2)2q

2u−1) . . . ,

we can regard the coefficients d (k) as certain formal series of elements of Uq(gl∞).


The following description of the algebra A0 is implied by Proposition 7.2.

Proposition 7.3. The elements d (1), d (2), . . . are algebraically independent and gener-

ate the algebra A0. �

We are now in a position to establish a relationship between the Olshanski algebra Amand the q-Yangian Yq(glm). It will be convenient to work with the subalgebra Yq(glm) of

Yq(glm) generated by the coefficients of the series

τij(u) = tij(u) t(0)jj , 1 6 i, j 6 m. (7.7)

As with the algebra Uq(glm), using the relations

qδja tia(u) t(0)jj = q

δij t(0)jj tia(u) and qδja tia(u) t

(0)jj = q

δij t(0)jj tia(u), (7.8)

it is easy to write down the defining relations of Yq(glm) in terms of the coefficients τ(r)ij of

the series τij(u).

Now we use the quantum Sylvester theorem (Theorem 6.6). Taking the composition of

the homomorphism (6.7) and the evaluation homomorphism (7.4), we obtain a homomor-

phism Yq(glm)→ Uq(gln) which can be written as

tij(u) 7→ qdet (T − T u−1)BiBj , (7.9)

where we use the quantum determinant of the submatrix of T − T u−1 corresponding tothe rows Bi and columns Bj, where Bi denotes the set {i,m + 1, . . . , n}. By the relations(5.30), the image of the series tij(u) under the homomorphism (7.9) commutes with the

elements tkl and tkl with m+ 1 6 k, l 6 n. Introduce the matrices T = [τij ] and T = [τij]with 1 6 i, j 6 n and set

qdet ′(T − T u−1)BiBj = qdet (T − T u−1)BiBj ∙ tjj tm+1,m+1 . . . tnn.

An explicit formula for this quantum minor in terms of τij and τij can be written with the

use of the parameter ind(σ) as in (7.6). Since the product tm+1,m+1 . . . tnn commutes with

qdet (T − T u−1)BiBj ∙ tjj (see (5.30)), the mapping

ϕn : τij(u) 7→ qdet′(T − T u−1)BiBj (7.10)

defines a homomorphism Yq(glm)→ Uq(gln). Furthermore, the same relation (5.30) impliesthat the image of the homomorphism (7.10) is contained in the centralizer Am(n) so that

we have a homomorphism

ϕn : Yq(glm)→ Am(n).


Using the explicit formula for qdet ′(T − T u−1)BiBj as in (7.6), we find that the diagram

Yq(glm) Yq(glm) ∙ ∙ ∙ Yq(glm) ∙ ∙ ∙

ϕm

y ϕm+1

y ϕn

y

Am(m) ←−−−om+1

Am(m+ 1) ←−−− ∙ ∙ ∙ ←−−−on

Am(n) ←−−−on+1

∙ ∙ ∙

is commutative. Note that the image of the generator τ(r)ij of Yq(glm) under ϕn has degree

6 r for any n. Hence the sequence of homomorphisms (ϕn | n > m) defines an algebra

homomorphism ϕ : Yq(glm) → Am which can be written in terms of the virtual quantumdeterminants by

ϕ : τij(u) 7→ qdet′(T − T u−1)BiBj ,

where Bi now denotes the infinite set {i,m+ 1,m+ 2, . . . }.

Theorem 7.4. The homomorphism ϕ is an algebra embedding of Yq(glm) into the

algebra Am.

Proof. We shall use a weak form of the Poincare–Birkhoff–Witt theorem for the al-

gebra Yq(glm) which can be verified by a direct argument; see e.g. [MRS, Lemma 3.2]

where a similar result is proved for a twisted version of the q-Yangian. By this theorem,

the ordered monomials in the generators τ(r)ij span the algebra Yq(glm). It will be suffi-

cient to prove that the images of these monomials under the homomorphism ϕ are linearly

independent.

As in Chapter 1, set A = C [q, q−1] and consider the A-subalgebra UA of Uq(gln) gener-ated by the elements τij and τij . Denote by Pn the algebra of polynomials in independentvariables xij with 1 6 i, j 6 n. By the Poincare–Birkhoff–Witt theorem for the algebra

Uq(gln), we have an isomorphism

UA ⊗A C ∼= Pn (7.11)

given by

τij 7→ xij for i > j and τij 7→ xij for i 6 j,

where the action of A on C is defined via the evaluation q = 1.Suppose now that the image under ϕ of a nontrivial linear combination of the ordered

monomials in the generators τ(r)ij is zero. We may assume that all coefficients of this linear

combination belong to A. Moreover, we may also assume that at least one coefficient doesnot vanish at q = 1 so that the image of this linear combination under the isomorphism

(7.11) yields a nontrivial linear combination of the corresponding elements of Pn. Observethat the image of the quantum determinant qdet ′(T −T u−1)BiBj occurring in (7.10) under

the isomorphism (7.11) coincides with the usual determinant det(X−Xu−1)BiBj , where X


denote the lower triangular matrix with entries xij for i > j and with all diagonal entries

equal to 1 while X is the upper triangular matrix with entries xij for i 6 j. Write

det(X −Xu−1)BiBj = λ(0)ij − λ

(1)ij u

−1 + ∙ ∙ ∙+ (−1)n−m+1λ(n−m+1)ij u−n+m−1.

The proof will be completed if we show that given any positive integer p, there exists a

sufficiently large value of n such that the polynomials λ(r)ij with 0 6 r 6 p and 1 6 i, j 6 m

(assuming i > j for r = 0) are algebraically independent. This will lead to a contradiction

with the assumption that the ordered monomials in the τ(r)ij are linearly dependent.

We shall be proving that the corresponding map

Λn : Cn2 → Cpm2+m(m−1)/2, (7.12)

given by

(xij | 1 6 i, j 6 n) 7→ (λ(r)kl | 1 6 k, l 6 m, 0 6 r 6 p)

with k > l for r = 0, is surjective.

Now we shall show that the map (7.12) is surjective for any n > (p + 1)m. In fact,we show that this map is surjective even when the variables xkl with m + 1 6 k 6 n are

specialized in the following way:

xkl = βkl, for l = 1, . . . ,m

and

xkl = δklαk, for l = m+ 1, . . . , n

where αm+1, . . . , αn are distinct complex numbers and the βkl are certain complex numbers

to be chosen below.

Under this specialization, for each fixed value of the index i ∈ {1, . . . ,m} consider thefollowing restriction of the map (7.12)

Λ(i)n : Cn → Cpm+i−1, (7.13)

given by

(xi1, xi2, . . . , xin) 7→

(λ(0)i1 , . . . , λ

(p)i1 , . . . , λ

(0)i,i−1, . . . , λ

(p)i,i−1, λ

(1)ii , . . . , λ

(p)ii , . . . , λ

(1)im, . . . , λ

(p)im),

so that the functions λ(r)ij are defined by the expansions

∣∣∣∣∣∣∣∣∣∣∣

txij xi,m+1 xi,m+2 ∙ ∙ ∙ xintβm+1,j t+ αm+1 0 ∙ ∙ ∙ 0

tβm+2,j 0 t+ αm+2 ∙ ∙ ∙ 0...

......

. . ....

tβn,j 0 0 ∙ ∙ ∙ t+ αn

∣∣∣∣∣∣∣∣∣∣∣

= tn−m+1λ(0)ij + ∙ ∙ ∙+ tλ

(n−m)ij ,


for j = 1, . . . , i− 1,

∣∣∣∣∣∣∣∣∣∣∣

t+ xii xi,m+1 xi,m+2 ∙ ∙ ∙ xintβm+1,i t+ αm+1 0 ∙ ∙ ∙ 0

tβm+2,i 0 t+ αm+2 ∙ ∙ ∙ 0...

......

. . ....

tβn,i 0 0 ∙ ∙ ∙ t+ αn

∣∣∣∣∣∣∣∣∣∣∣

= tn−m+1 + tn−mλ(1)ii + ∙ ∙ ∙+ λ

(n−m+1)ii ,

and∣∣∣∣∣∣∣∣∣∣∣

xij xi,m+1 xi,m+2 ∙ ∙ ∙ xintβm+1,j t+ αm+1 0 ∙ ∙ ∙ 0

tβm+2,j 0 t+ αm+2 ∙ ∙ ∙ 0...

......

. . ....

tβn,j 0 0 ∙ ∙ ∙ t+ αn

∣∣∣∣∣∣∣∣∣∣∣

= tn−mλ(1)ij + ∙ ∙ ∙+ λ

(n−m+1)ij ,

for j = i+ 1, . . . ,m. More explicitly, we have

λ(r)ij = xij er(αm+1, . . . , αn)−

n∑

k=m+1

βkjxik er−1(αm+1, . . . , αk, . . . , αn)

for j = 1, . . . , i− 1 and r > 0,

λ(r)ii = xiier−1(αm+1, . . . , αn)

−n∑

k=m+1

βkixik er−1(αm+1, . . . , αk, . . . , αn) + er(αm+1, . . . , αn)

for r > 1, and

λ(r)ij = xij er−1(αm+1, . . . , αn)−

n∑

k=m+1

βkjxik er−1(αm+1, . . . , αk, . . . , αn)

for j = i+ 1, . . . ,m and r > 1.Here, as in Chapter 4, for any r > 1, er(z1, . . . , zl) denotes the r-th elementary sym-

metric function in variables z1, . . . , zl with l > r.

Since the λ(r)ij are linear functions in the variables xi1, . . . , xin, in order to establish the

surjectivity of the map (7.13), it will be sufficient to demonstrate that the rank of the

corresponding coefficient matrix is maximal, that is, equal to pm + i − 1. Writing downthe matrix in an explicit form and using the observation that λ

(0)ij = xij for j = 1, . . . , i− 1


we conclude that the claim will follow by proving that the matrix

−βm+1,1e0,m+1 ∙ ∙ ∙ −βn1e0,n−βm+1,1e1,m+1 ∙ ∙ ∙ −βn1e1,n

... ∙ ∙ ∙...

−βm+1,1ep−1,m+1 ∙ ∙ ∙ −βn1ep−1,n... ∙ ∙ ∙

...... ∙ ∙ ∙

...

−βm+1,me0,m+1 ∙ ∙ ∙ −βnme0,n−βm+1,me1,m+1 ∙ ∙ ∙ −βnme1,n

... ∙ ∙ ∙...

−βm+1,mep−1,m+1 ∙ ∙ ∙ −βnmep−1,n

(7.14)

has rank pm, where erk = er(αm+1, . . . , αk, . . . , αn). Now, we use the assumption n−m >pm and Lemma 4.10. Giving appropriate values to the parameters βkl, we may choose a

submatrix of (7.14) of size pm × pm of the form (4.9), where all the p × p blocks Ei arenon-singular and the parameters βi are distinct. Thus, the claim follows by the application

of Lemma 4.11.

Finally, by allowing the index i in (7.13) to vary and by choosing the parameters βkl in

the same way for each value of i ∈ {1, . . . ,m}, we get a surjective map

Cmn → Cpm2+m(m−1)/2,

given by

(xij | 1 6 i 6 m, 1 6 j 6 n) 7→ (λ(r)il | 1 6 i, l 6 m, 0 6 r 6 p)

with i > l for r = 0, thus completing the proof. �

The argument used in the proof of Theorem 7.4 provides a proof of the Poincare–

Birkhoff–Witt theorem for the algebra Yq(glm) and hence for Yq(glm).

Corollary 7.5. Given any ordering on the set of generators of the algebra Yq(glm),

the ordered monomials in the generators form a basis of Yq(glm).

Proof. As we showed in the proof of Theorem 7.4, given any finite family of ordered

monomials in the generators τ(r)ij of the algebra Yq(glm), we can choose a sufficiently large

value of n such that the images of the monomials under the homomorphism (7.10) are

linearly independent. As the ordered monomials span Yq(glm), we may conclude that they

form a basis of Yq(glm). The corresponding statement for the algebra Yq(glm) is now

immediate from the relations (7.7) and (7.8). �

We believe an analogue of the tensor product decomposition (0.4) for the algebra Amtakes place. Namely, denote by A0 the commutative subalgebra of Am generated by the


coefficients of the virtual quantum determinant qdet ′(T −T u−1)BB with B = {m+1,m+2, . . . }.

Conjecture 7.6. We have the tensor product decomposition

Am = A0 ⊗ Yq(glm), (7.15)

where the Yangian Yq(glm) is identified with its image under the embedding ϕ.

CHAPTER 8

Skew representations of Y(gln)

In this chapter and the next we study aspects of the representation theory of Y(gln)

and Uq(gln).

We will employ the dual form tij(u) 7→ t [ij(u) of the homomorphism Y(gln)→ Y(glm+n)defined in (3.13). More precisely, for m > 0, consider the homomorphism Y(gln) →Y(glm+n) given by

tij(u) 7→(t 1...m1...m(u)

)−1∙ t 1...m,m+i1...m,m+j(u);

see (1.20) and Corollary 3.4. Its composition with the evaluation homomorphism (1.17) is

a homomorphism Y(glm)→ U(glm+n) which can be given by

tij(u) 7→ qdet(1 + E u−1

)−1AA∙ qdet

(1 + E u−1

)AiAj

, (8.1)

where A = {1, . . . ,m} and Ai = {1, . . . ,m,m+ i}. By Lemma 3.1, the image of t(1)ij under

the homomorphism (8.1) is Em+i,m+j. Therefore, Corollary 2.8 implies that the image of

this homomorphism is contained in the centralizer U(glm+n)glm , where glm is regarded as

a natural subalgebra of glm+n.

The irreducible highest weight representation L(ν(u) of the Yangian Y(gln) with the

highest weight ν(u) is generated by a non-zero vector ξ such that

tij(u) ζ = 0, for 1 6 i < j 6 n,

tii(u) ζ = νi(u) ζ, for 1 6 i 6 n,

where ν(u) = (ν1(u), . . . , νn(u)) is a certain n-tuples of formal power series in u−1 with

constant term 1.

Suppose that there exist polynomials P1(u), . . . , Pn−1(u) in u, all with constant term

1, such thatνk(u)

νk+1(u)=Pk(u+ 1)

Pk(u)(8.2)

for any k = 1, . . . , n − 1. Then the representation L(ν(u)) is finite-dimensional and wecall the Pk(u) the Drinfeld polynomials of this representation. Furthermore, given an n-

tuple (P1(u), . . . , Pn(u)) of monic polynomials there exists a finite-dimensional irreducible

representation having this n-tuple as the family of its Drinfeld polynomials.

Let λ = (λ1, . . . , λn) be an n-tuple of integers such that λ1 > λ2 > ∙ ∙ ∙ > λn. Let L(λ)

be the irreducible highest weight module of gln with highest weight λ.

63

64 8. SKEW REPRESENTATIONS OF Y(glN )

A pattern Λ (associated with λ) is a sequence of rows of integers Λn,Λn−1, . . . ,Λ1, where

Λk = (λk1, . . . , λkk) is the k-th row from the bottom, the top row Λn coincides with λ, and

the following betweenness conditions are satisfied: for k = 1, . . . , n− 1

λk+1,i+1 6 λki 6 λk+1,i for i = 1, . . . , k. (8.3)

We shall be using the notation lki = λki − i + 1. There exists a basis {ξΛ} in L(λ)parameterized by the patterns Λ such that the action of the generators of U(gln) is given

by

Ekk ξΛ = wk ξΛ, wk =k∑

i=1

λki −k−1∑

i=1

λk−1,i, (8.4)

Ek,k+1 ξΛ = −k∑

j=1

( lk+1,1 − lkj) ∙ ∙ ∙ ( lk+1,k+1 − lkj)( lk1 − lkj) ∙ ∙ ∙ ∧j ∙ ∙ ∙ ( lkk − lkj)

ξΛ+δkj , (8.5)

Ek+1,k ξΛ =k∑

j=1

( lk−1,1 − lkj) ∙ ∙ ∙ ( lk−1,k−1 − lkj)( lk1 − lkj) ∙ ∙ ∙ ∧j ∙ ∙ ∙ ( lkk − lkj)

ξΛ−δkj , (8.6)

where Λ ± δkj is obtained from Λ by replacing the entry λkj with λkj ± 1, and ξΛ issupposed to be equal to zero if Λ is not a pattern; the symbol ∧j indicates that the j-thfactor is skipped. This is the Gelfand–Tsetlin basis for L(λ), originally due to I. Gelfand

and M. Tsetlin [GT].

The representation L(λ) of gln can be extended to a representation of Y(gln) via the

evaluation homomorphism (1.17). This Y(gln)-module is called the evaluation module. Its

Drinfeld polynomials are given by

Pk(u) = (u+ λk+1)(u+ λk+1 + 1) ∙ ∙ ∙ (u+ λk − 1), (8.7)

for k = 1, . . . , n− 1.Consider now a glm+n-highest weight λ = (λ1, . . . , λm+n) so that the λi are complex

numbers satisfying the condition λi − λi+1 ∈ Z+ for all values of the index i = 1, . . . ,m +n − 1. Furthermore, let μ = (μ1, . . . , μm) be a glm-highest weight. The vector spaceHomglm

(L(μ), L(λ)

)is isomorphic to the subspace L(λ)+μ of L(λ) which consists of glm-

highest vectors of weight μ,

L(λ)+μ = {η ∈ L(λ) | Eij η = 0 1 6 i < j 6 m and

Eiiη = μiη i = 1, . . . ,m}.

The subspace L(λ)+μ is nonzero if and only if

λi − μi ∈ Z+ and μi − λi+n ∈ Z+ for i = 1, . . . ,m. (8.8)

This follows easily from the properties of the Gelfand–Tsetlin basis of L(λ). We shall

suppose that the conditions (8.8) hold. Observe that L(λ)+μ is a natural module over the

8. SKEW REPRESENTATIONS OF Y(glN ) 65

centralizer U(glm+n)glm ; see the comment at the start of the chapter. We now make L(λ)+μ

into a module over the Yangian Y(gln) via the homomorphism (8.1).

Proposition 8.1. The Y(gln)-module L(λ)+μ is irreducible.

Proof. It is well-known that L(λ)+μ is an irreducible representation of the centralizer

U(glm+n)glm . Since the elements of the center of U(glm) act on L(λ)

+μ as scalar operators,

the claim will follow if we show that the centralizer is generated by the image of the

homomorphism (8.1) and the center of U(glm). However, the center of U(glm) is generated

by the coefficients of the series qdet (1 + E u−1)AA. Therefore, the obvious modification of

Theorem 2.11 implies that the coefficients of the series

qdet (1 + E u−1)AA and qdet (1 + E u−1)AiAj , 1 6 i, j 6 n,

generate the centralizer U(glm+n)glm ; cf. the arguments used in Chapter 7 for the proof of

Theorem 7.4. This implies the statement. �

The subspace L(λ)+μ admits a basis {ζΛ} labelled by the trapezium Gelfand–Tsetlinpatterns Λ of the form

λ1 λ2 λ3 ∙ ∙ ∙ λm+n−1 λm+n

λm+n−1,1 λm+n−1,2 ∙ ∙ ∙ λm+n−1,m+n−1∙∙∙

∙∙∙

∙ ∙ ∙∙∙∙

λm+1,1 λm+1,2 ∙ ∙ ∙ λm+1,m+1

μ1 μ2 ∙ ∙ ∙ μm

These arrays are formed by complex numbers λki satisfying the betweenness conditions

λki − λk+1,i+1 ∈ Z+ and λk+1,i − λki ∈ Z+ (8.9)

for k = m,m+ 1, . . . ,m + n− 1 and 1 6 i 6 k, where we have set

λmi = μi, i = 1, . . . ,m and λm+n,j = λj, j = 1, . . . ,m + n.

note that for fixed λ and μ all entries of Λ belong to the same Z-coset of C . Ordering theelements of the coset by their real parts we can rewrite (8.9) in an equivalent form as

λk+1,i+1 6 λki 6 λk+1,i.

Using this convention, introduce the trapezium array Λ0 with the entries given by

λ0m+k,i = min{λi, μi−k}, k = 1, . . . , n− 1, i = 1, . . . ,m + k, (8.10)

where we assume that μj is sufficiently large if j 6 0. It is easy to check that Λ0 is apattern.


Proposition 8.2. The basis vector ζΛ0 is the highest vector of the Y(gln)-module

L(λ)+μ .

Proof. The vector space L(λ)+μ is a natural module over the subalgebra U(gln) of

Y(gln). The action of gln on the basis vectors is given by (8.4), (8.5) and (8.6), where

gln is identified with the subalgebra of glm+n spanned by the elements Em+i,m+j with

1 6 i, j 6 n. In particular, we see that the gln-weight w of ζΛ0 ,

w = (w1, . . . , wn), wp =

m+p∑

i=1

λ0m+p,i −m+p−1∑

i=1

λ0m+p−1,i

is a maximal with respect to the standard partial ordering on the set of weights of the

gln-module L(λ)+μ . Hence the vector ζΛ0 is annihilated by tij(u) with 1 6 i < j 6 n. The

proof is now completed by the application of Proposition 8.1 and by observation that the

subspace of highest weight vectors is one-dimensional. �

Given three elements i, j, k of a Z-coset in C , we shall denote by middle{i, j, k} that ofthe three which is between the two others.

Theorem 8.3. The Y(gln)-module L(λ)+μ is isomorphic to the highest weight represen-

tation L(ν(u)), where the components of the highest weight ν(u) are found by

νk(u) =(u+ ν

(1)k )(u+ ν

(2)k − 1) . . . (u+ ν

(m+1)k −m)

(u+ μ1)(u+ μ2 − 1) . . . (u+ μm −m+ 1)(u−m)

where k = 1, . . . , n and

ν(i)k = middle{μi−1, μi, λk+i−1}, (8.11)

assuming μ0 is sufficiently large and μm+1 is sufficiently small.

Proof. By Propositions 8.1 and 8.2, we only need to calculate the highest weight of

the Y(gln)-module L(λ)+μ . As ζΛ0 is the highest vector, by (2.17) we have for any 1 6 k 6 n

t 1... k1... k (u) ζΛ0 = ν1(u) ν2(u− 1) . . . νk(u− k + 1) ζΛ0 .

On the other hand, by Lemma 3.3, the action of this quantum minor can also be found by

applying the evaluation homomorphism (1.17) to the following series with coefficients in

Y(glm+n), (t 1...m1...m (u)

)−1∙ t 1...m,m+1...m+k1...m,m+1...m+k (u), (8.12)

and then applying the image to the vector ζΛ0 . We obtain the relation

ν1(u) ν2(u− 1) . . . νk(u− k + 1)

=

(1 + λ0m+k,1u

−1). . .(1 + λ0m+k,m+k(u−m− k + 1)

−1)

(1 + μ1u−1

). . .(1 + μm(u−m+ 1)−1

) .

8. SKEW REPRESENTATIONS OF Y(glN ) 67

Replacing here k by k − 1 we derive a formula for νk(u− k + 1) which implies

νk(u) =

(1 + λ0m+k,1(u+ k − 1)

−1). . .(1 + λ0m+k,m+k (u−m)

−1)

(1 + λ0m+k−1,1(u+ k − 1)

−1). . .(1 + λ0m+k−1,m+k−1(u−m+ 1)

−1) .

note that

λ0m+k,i = λ0m+k−1,i = λi for i = 1, . . . , k − 1

and1 + λ0m+k,k+j−1(u− j + 1)

−1

1 + λ0m+k−1,k+j−1(u− j + 1)−1=1 + ν

(j)k (u− j + 1)

−1

1 + μj(u− j + 1)−1

for j = 1, . . . ,m. Since λ0m+k,m+k = ν(m+1)k this gives the desired formula. �

In the case where the components of λ and μ are nonnegative integers, we may regard

them as partitions. The first condition in (8.8) implies that the diagram of μ is contained

in the diagram of λ. The skew diagram λ/μ is the set-theoretical difference of the diagrams

of λ and μ. By the content of a cell α ∈ λ/μ we mean the number c(α) = j − i if α occursin row i and column j.

Corollary 8.4. The Drinfeld polynomials P1(u), . . . , Pn−1(u) of the representation

L(λ)+μ of Y(gln) are given by

Pk(u) =m+1∏

i=1

(u+ ν(i)k+1 − i+ 1)(u+ ν

(i)k+1 − i+ 2) . . . (u+ ν

(i)k − i),

for k = 1, . . . , n− 1. If λ and μ are partitions then the formula can also be written as

Pk(u) =∏

α

(u+ c(α)

),

where α runs over the top boxes of the columns of height k in the diagram of λ/μ.

Proof. The formulae follow from Theorem 8.3 and the relations between the highest

weight and Drinfeld polynomials of a representation of Y(gln); see (8.2) and Theorem 8.3.

�

Example 8.5. If λ = (10, 8, 5, 4, 2) and μ = (6, 3) then the corresponding skew diagram

is

The Drinfeld polynomials of the Y(gl3)-module L(λ)+μ are

P1(u) = (u+ 4)(u+ 8)(u+ 9), P2(u) = u(u+ 3)(u+ 6)(u+ 7). �


Note that in the case m = 0 the skew representation L(λ)+μ of Y(gln) coincides with

the evaluation module L(λ). Clearly, in this case the Drinfeld polynomials provided by

Corollary 8.4 coincide with those given by (8.7).

8.1. Gelfand–Tsetlin characters

For a given n consider the set Pn of tuples λ(u) =(λ1(u), . . . , λn(u)

)where each λi(u)

is a formal series in u−1. Then Pn is an abelian group with respect to the component-wisemultiplication of the tuples. Consider the group ring Z [Pn] of the abelian group Pn whoseelements are finite linear combinations of the form

∑mλ(u)[λ(u)], where mλ(u) ∈ Z .

Let us introduce the series hi(u) with coefficients in Y(gln) by the formulae

hi(u) = t1... i−11... i−1(u− i+ 1)

−1 t 1... i1... i(u− i+ 1), i = 1, . . . , n.

The Gelfand–Tsetlin subalgebra Hn is the subalgebra of Y(gln) generated by the coefficients

h(r)i of the series h1(u), . . . , hn(u). Due to (2.21), it is commutative subalgebra (in fact it

is a maximal commutative subalgebra).

Definition 8.6. Suppose that V is a finite-dimensional representation of the Yangian

Y(gln). For any λ(u) ∈ Pn, the corresponding Gelfand–Tsetlin subspace Vλ(u) consists ofthe vectors v ∈ V with the property that for each i = 1, . . . , n and each r > 1 there existsp > 1 such that

(h(r)i − λ

(r)i

)pv = 0. Then the Gelfand–Tsetlin character of V is the

element of Z [Pn] defined by

chV =∑

λ(u)∈Pn

(dimVλ(u)

)[λ(u)].

�

We conclude this chapter with a calculation of the Gelfand–Tsetlin character of the skew

representation L(λ)+μ of the Yangian. It follows from Theorem 8.3 that the composition

of the Y(gln)-module L(λ)+μ with a shift automorphism (1.20) is isomorphic to the skew

representation L(λ′)+μ′ with the shifted components of λ and μ,

λ′i = λi − c, μ′i = μi − c.

So we may assume without loss of generality that λ and μ are partitions. The formulae in

the general case can then be obtained by applying an appropriate shift in u.

Introduce the following special elements of the group ring Z [Pn] by

xi,a =[(1, . . . , 1 + (u+ a+ i− 1)−1, . . . , 1

)], 1 6 i 6 n, a ∈ C ,

where the only non-unit series is the i-th component of the tuple.

A semistandard λ/μ-tableau T is obtained by writing the numbers 1, . . . , n into thecells of the skew diagram λ/μ in such a way that the elements in each row weakly increase

while the elements in each column strictly increase. By T (α) we denote the entry of T inthe cell α ∈ λ/μ.

8.1. GELFAND–TSETLIN CHARACTERS 69

Corollary 8.7. The Gelfand–Tsetlin character of the Y(gln)-module L(λ)+μ is given

by

chL(λ)+μ =∑

T

∏

α∈λ/μ

xT (α),c(α),

summed over all semistandard λ/μ-tableaux T . In particular, in the case m = 0 this givesthe Gelfand–Tsetlin character of the evaluation module for Y(gln).

Proof. We start with the particular case m = 0 so that L(λ)+μ coincides with the

evaluation module L(λ). The coefficients of the quantum determinant of Y(gln) act on

L(λ) as scalar operators, so that

qdetT (u)|L(λ) =(1 + λ1u

−1). . .(1 + λn (u− n+ 1)

−1).

Observe that, regarding λ as a diagram, we can write the product here as∏

α∈λ

u+ c(α) + 1

u+ c(α).

Also note that we have the decomposition

qdetT (u) = h1(u)h2(u− 1) . . . hn(u− n+ 1).

now we employ the well-known bijection between the patterns associated with λ and the

semistandard λ-tableaux; namely, the pattern Λ can be viewed as the sequence of diagrams

Λ1 ⊂ Λ2 ⊂ ∙ ∙ ∙ ⊂ Λn = λ,

where the Λk = (λk1, . . . , λkk) are the rows of Λ. The corresponding semistandard λ-tableau

is obtained by placing the entry k into each box of Λk/Λk−1. Hence, using the properties

of the Gelfand–Tsetlin basis of L(λ), for any basis vector ζΛ ∈ L(λ) and any 1 6 k 6 n we

have

h1(u)h2(u− 1) . . . hk(u− k + 1) ζΛ =∏

α∈Λk

u+ c(α) + 1

u+ c(α)ζΛ.

This implies

hk(u− k + 1) ζΛ =∏

α∈Λk/Λk−1

u+ c(α) + 1

u+ c(α)ζΛ. (8.13)

Thus, the element of Z [Pn] corresponding to the eigenvalue of hk(u) on ζΛ coincides withthe product ∏

α∈Λk/Λk−1

xk,c(α).

This completes the proof in the case m = 0.

In the skew case (m > 1) we note that the action of the quantum minor t 1... k1... k (u) on the

basis vector ζΛ ∈ L(λ)+μ is found by the application of the evaluation homomorphism (1.17)

to the series (8.12). Hence the formula (8.13) remains valid in the skew case as well, where


Λk now denotes the row (λm+k,1, . . . , λm+k,m+k) of the trapezium pattern Λ and we use the

natural bijection between the trapezium patterns and the semistandard λ/μ-tableaux. �

CHAPTER 9

Skew representations of Uq(gln)

Consider the quantum affine algebra Uq(gln) and suppose, as before, that the complex

parameter q is nonzero and not a root of unity. The irreducible (pseudo) highest weight

representation L(ν(u), ν(u)) of Uq(gln) with the (pseudo) highest weight (ν(u), ν(u)) is

generated by a nonzero vector ζ such that

tij(u) ζ = 0, tij(u) ζ = 0 for 1 6 i < j 6 n,

tii(u) ζ = νi(u) ζ, tii(u) ζ = νi(u) ζ for 1 6 i 6 n,

where ν(u) = (ν1(u), . . . , νn(u)) and ν(u) = (ν1(u), . . . , νn(u)) are certain n-tuples of formal

power series in u−1 and u, respectively.

Suppose that there exist polynomials P1(u), . . . , Pn−1(u) in u, all with constant term

1, such thatνk(u)

νk+1(u)= q− degPk ∙

Pk(uq2)

Pk(u)=

νk(u)

νk+1(u)(9.1)

for any k = 1, . . . , n−1. The first equality in (9.1) is understood in the sense that the ratioof polynomials has to be expanded as a power series in u−1, while for the second equality the

same ratio has to be expanded as a power series in u. Then the representation L(ν(u), ν(u))

is finite-dimensional. The Pk(u) are called the Drinfeld polynomials of this representation.

The families of Drinfeld polynomials parameterize the type 1 finite-dimensional irreducible

representations of the subalgebra Uq(sln) of Uq(gln); see [CP, Section 12], [DF], [FM].

Let λ = (λ1, . . . , λn) be an n-tuple of integers such that λ1 > ∙ ∙ ∙ > λn. The cor-

responding irreducible highest weight representation L(λ) of Uq(gln) is generated by a

nonzero vector ξ such that

tij ξ = 0 for 1 6 i < j 6 n,

tii ξ = qλi ξ for 1 6 i 6 n.

This representation is a q-analogue of the irreducible gln-module with the highest weight

λ. In particular, these modules have the same dimension.

We shall need an analogue of the Gelfand–Tsetlin basis for the module L(λ) [J2]; see

also [KS, Section 7.3.3] for more details. As in Chapter 8, a pattern Λ (associated with

λ) is a sequence of rows of integers Λn,Λn−1, . . . ,Λ1, where Λk = (λk1, . . . , λkk) is the k-th

row from the bottom, the top row Λn coincides with λ, and the following betweenness

71

72 9. SKEW REPRESENTATIONS OF Uq(glN )

conditions are satisfied: for k = 1, . . . , n− 1

λk+1,i+1 6 λki 6 λk+1,i for i = 1, . . . , k. (9.2)

We shall be using the notation lki = λki − i+ 1. Also, for any integer m we set

[m] =qm − q−m

q − q−1. (9.3)

There exists a basis {ξΛ} in L(λ) parameterized by the patterns Λ such that the action ofthe generators of Uq(gln) is given by

tkk ξΛ = qwk ξΛ, wk =

k∑

i=1

λki −k−1∑

i=1

λk−1,i, (9.4)

ek ξΛ = −k∑

j=1

[ lk+1,1 − lkj ] ∙ ∙ ∙ [ lk+1,k+1 − lkj ][ lk1 − lkj] ∙ ∙ ∙ ∧j ∙ ∙ ∙ [ lkk − lkj ]

ξΛ+δkj , (9.5)

fk ξΛ =k∑

j=1

[ lk−1,1 − lkj] ∙ ∙ ∙ [ lk−1,k−1 − lkj][ lk1 − lkj ] ∙ ∙ ∙ ∧j ∙ ∙ ∙ [ lkk − lkj]

ξΛ−δkj , (9.6)

where Λ±δkj is obtained from Λ by replacing the entry λkj with λkj±1, and ξΛ is supposedto be equal to zero if Λ is not a pattern; the symbol ∧j indicates that the j-th factor isskipped.

The representation L(λ) of Uq(gln) can be extended to a representation of Uq(gln) via

the evaluation homomorphism; see (1.45). This Uq(gln)-module is called the evaluation

module. Its Drinfeld polynomials are given by

Pk(u) = (1− q2λk+1u)(1− q2λk+1+2u) . . . (1− q2λk−2u), (9.7)

for k = 1, . . . , n − 1. We shall be using some formulae for the action of certain quantumminors of the matrices T (u) and T (u) in the Gelfand–Tsetlin basis of L(λ). Set

Tij(u) =u tij − u−1 tijq − q−1

. (9.8)

Clearly, (q− q−1)Tij(u) equals u times the image of tij(u2) under the evaluation homomor-phism (1.45). We also define the corresponding quantum minors T a1∙∙∙ ar

b1∙∙∙ br (u) by using the

formulae (5.23) or (5.25), so that, in particular, if a1 < ∙ ∙ ∙ < ar then

T a1∙∙∙ arb1∙∙∙ br (u) =

∑

σ∈Sr

(−q)−l(σ) ∙ Taσ(1)b1(u) ∙ ∙ ∙Taσ(r)br(q−r+1u). (9.9)

For any k = 1, . . . , n we have in L(λ),

T 1∙∙∙ k1∙∙∙ k(u) ξΛ =k∏

i=1

uq lki − u−1q−lki

q − q−1ξΛ. (9.10)

9. SKEW REPRESENTATIONS OF Uq(glN ) 73

Indeed, the coefficients of the polynomial T 1∙∙∙ k1∙∙∙ k(u) commute with the elements of the

subalgebra Uq(glk) and so the relation is verified by the application of the polynomial to

the highest vector of the Uq(glk)-module L(Λk). Furthermore, for any k = 1, . . . , n−1 andj = 1, . . . , k we have

T 1∙∙∙ k1∙∙∙ k−1, k+1(q−lkj) ξΛ = −[ lk+1,1 − lkj ] ∙ ∙ ∙ [ lk+1,k+1 − lkj ] ξΛ+δkj (9.11)

and

T 1∙∙∙ k−1, k+11∙∙∙ k (q−lkj) ξΛ = [ lk−1,1 − lkj] ∙ ∙ ∙ [ lk−1,k−1 − lkj ] ξΛ−δkj . (9.12)

The formulae (9.11) and (9.12) appeared for the first time in [NT1] in a slightly different

form. For the proof of (9.11), it suffices to use the relation

T 1∙∙∙ k1∙∙∙ k−1, k+1(u) = T1∙∙∙ k1∙∙∙ k(u) ek − q ek T

1∙∙∙ k1∙∙∙ k(u),

which can be deduced from (2.4), and then apply (9.5) and (9.10). The proof of (9.12) is

similar with the use of (9.6) and (9.10).

Suppose now that m is a non-negative integer and let λ = (λ1, . . . , λm+n) be an (m+n)-

tuple of integers such that λ1 > ∙ ∙ ∙ > λm+n. Furthermore, let μ = (μ1, . . . , μm) be an

m-tuple of integers such that μ1 > ∙ ∙ ∙ > μm. Regarding Uq(glm) as a natural subalgebra

of Uq(glm+n), consider the space HomUq(glm)(L(μ), L(λ)

). This vector space is isomorphic

to the subspace L(λ)+μ of L(λ) which consists of Uq(glm)-singular vectors of weight μ,

L(λ)+μ = {η ∈ L(λ) | tij η = 0 1 6 i < j 6 m and

tiiη = qμi η i = 1, . . . ,m}.

Note that L(λ)+μ is nonzero if and only if

λi > μi > λi+n for i = 1, . . . ,m;

see e.g. [CP]. We shall assume that these inequalities hold. In that case, L(λ)+μ admits a

basis {ζΛ} labelled by the trapezium Gelfand–Tsetlin patterns Λ of the form

λ1 λ2 λ3 ∙ ∙ ∙ λm+n−1 λm+n

λm+n−1,1 λm+n−1,2 ∙ ∙ ∙ λm+n−1,m+n−1∙∙∙

∙∙∙

∙ ∙ ∙∙∙∙

λm+1,1 λm+1,2 ∙ ∙ ∙ λm+1,m+1

μ1 μ2 ∙ ∙ ∙ μm

These arrays are formed by integers λki satisfying the betweenness conditions

λk+1,i+1 6 λki 6 λk+1,i


for k = m,m+ 1, . . . ,m + n− 1 and 1 6 i 6 k, where we have set

λmi = μi, i = 1, . . . ,m and λm+n,j = λj, j = 1, . . . ,m + n.

Observe that L(λ)+μ is a natural module over the subalgebra U′q(gln) of Uq(glm+n)

generated by the elements tij and tij with m + 1 6 i, j 6 m + n. Consider the trapezium

pattern Λ0 with the entries given by

λ0m+k,i = min{λi, μi−k}, k = 1, . . . , n− 1, i = 1, . . . ,m + k, (9.13)

where we assume μj = +∞ if j 6 0. Then

tpp ζΛ0 = qwp ζΛ0 , wp =

p∑

i=1

λ0pi −p−1∑

i=1

λ0p−1,i, p = m+ 1, . . . ,m + n.

The weight (wm+1, . . . , wm+n) of ζΛ0 is maximal with respect to the standard ordering on

the set of weights of the U′q(gln)-module L(λ)+μ .

We now make L(λ)+μ into a module over the quantum affine algebra Uq(gln) by tak-

ing the composition of the homomorphism (6.12) with the evaluation homomorphism

Uq(glm+n) → Uq(glm+n) given by (1.45). Thus, we can write the formulae for the ac-tion of Uq(gln) on L(λ)

+μ in the form

tij(u) 7→(qdet (T − T u−1)AA

)−1∙ qdet (T − T u−1)AiAj ,

tij(u) 7→(qdet (T − T u)AA

)−1∙ qdet (T − T u)AiAj ,

where A = {1, . . . ,m} and Ai = {1, . . . ,m,m + i} while T and T denote the generatormatrices for the algebra Uq(glm+n). The subspace L(λ)

+μ of L(λ) is preserved by these

operators due to relations (5.30).

Theorem 9.1. The representation L(λ)+μ of Uq(gln) is irreducible.

Proof. By Corollary 6.10, for the action of quantum minors on L(λ)+μ we can write

t a1∙∙∙ arb1∙∙∙ br (u2) 7→ T 1∙∙∙m1∙∙∙m(u)

−1 T 1∙∙∙m,m+a1∙∙∙m+ar1∙∙∙m,m+b1∙∙∙m+br (u)r∏

i=1

q − q−1

u q−m−i+1,

where ai, bi ∈ {1, . . . , n}. Note that the coefficients of the polynomial T1∙∙∙m1∙∙∙m(u) act on

L(λ)+μ as scalar operators found from

T 1∙∙∙m1∙∙∙m(u) ζΛ =m∏

i=1

uq ì − u−1q−ì

q − q−1ζΛ, ì = μi − i+ 1.

On the other hand, by the formulae (9.11) and (9.12), each of the operators

T 1∙∙∙m,m+1∙∙∙m+k1∙∙∙m,m+k∙∙∙m+k−1,m+k+1(q−lm+k,i) and T 1∙∙∙m,m+1∙∙∙m+k−1,m+k+11∙∙∙m,m+k∙∙∙m+k (q−lm+k,i)

9. SKEW REPRESENTATIONS OF Uq(glN ) 75

takes a basis vector ζΛ of L(λ)+μ to another basis vector with a nonzero coefficient provided

the respective array Λ + δm+k,i or Λ− δm+k,i is a pattern. �

Remark 9.2. The above proof actually shows that L(λ)+μ is irreducible as a represen-

tation of the q-Yangian Yq(gln).

Moreover, as the coefficients of the quantum minors

t 1∙∙∙ r1∙∙∙ r(u), t 1∙∙∙ r1∙∙∙ r−1,r+1(u), t 1∙∙∙ r−1,r+11∙∙∙ r (u)

and

t1∙∙∙ r1∙∙∙ r(u), t

1∙∙∙ r1∙∙∙ r−1,r+1(u), t

1∙∙∙ r−1,r+11∙∙∙ r (u)

with r > 1 generate the algebra Uq(gln), Corollary 6.10 together with the formulae (9.10),(9.11) and (9.12) provide an explicit realization of the Uq(gln)-module L(λ)

+μ . �

Our next goal is to calculate the highest weight and Drinfeld polynomials of the Uq(gln)-

module L(λ)+μ . As we have noticed above, ζΛ0 is a unique vector of maximal weight with

respect to the standard ordering on the set of weights of the U′q(gln)-module L(λ)+μ . By

the defining relations of Uq(gln), we have

t(0)jj tia(u) = q

δij−δja tia(u) t(0)jj and t

(0)jj tia(u) = q

δij−δja tia(u) t(0)jj .

This implies that ζΛ0 is the highest vector of the Uq(gln)-module L(λ)+μ . The following

theorem provides an identification of this module; cf. [Mo], [NT2]. Given three integers

i, j, k we shall denote by middle{i, j, k} that of the three which is between the two others.

Theorem 9.3. The Uq(gln)-module L(λ)+μ is isomorphic to the highest weight repre-

sentation L(ν(u), ν(u)), where the components of the highest weight are found by

νk(u) =(q ν

(1)k − q−ν

(1)k u−1) . . . (q ν

(m+1)k − q−ν

(m+1)k +2mu−1)

(q μ1 − q−μ1 u−1) . . . (q μm − q−μm+2m−2u−1)

and

νk(u) =(q−ν

(1)k − q ν

(1)k u) . . . (q−ν

(m+1)k − q ν

(m+1)k −2mu)

(q−μ1 − q μ1 u) . . . (q−μm − q μm−2m+2u),

where k = 1, . . . , n and

ν(i)k = middle{μi−1, μi, λk+i−1} (9.14)

assuming μm+1 = −∞, and μ0 = +∞.

Proof. Due to Theorem 9.1, we only need to calculate the highest weight of the

Uq(gln)-module L(λ)+μ . As ζΛ0 is the highest vector, we have for any 1 6 k 6 n

t 1∙∙∙ k1∙∙∙ k(u) ζΛ0 = ν1(u) ν2(q−2u) . . . νk(q

−2k+2u) ζΛ0 .


On the other hand, by Corollary 6.10, the action of this quantum minor can be found by

applying the evaluation homomorphism (1.45) to the following series with coefficients in

Uq(glm+n),(t 1...m1...m(u)

)−1∙ t 1...m,m+1...m+k1...m,m+1...m+k(u),

and then applying the image to the vector ζΛ0 . This yields the relation

ν1(u) ν2(q−2u) . . . νk(q

−2k+2u)

=(q λ

(0)m+k,1 − q−λ

(0)m+k,1u−1) . . . (q λ

(0)m+k,m+k − q−λ

(0)m+k,m+k+2m+2k−2u−1)

(q μ1 − q−μ1u−1) . . . (q μm − q−μm+2m−2u−1).

Hence, replacing here k by k − 1 and u by q2k−2u, we get

νk(u) =

(q λ(0)m+k,1 − q−λ

(0)m+k,1−2k+2u−1) . . . (q λ

(0)m+k,m+k − q−λ

(0)m+k,m+k+2mu−1)

(q λ(0)m+k−1,1 − q−λ

(0)m+k−1,1−2k+2u−1) . . . (q λ

(0)m+k−1,m+k−1 − q−λ

(0)m+k−1,m+k−1+2m−2u−1)

.

By the definition (9.13) of the pattern Λ0, we have

λ(0)m+k,i = λ

(0)m+k−1,i = λi for i = 1, . . . , k − 1

while for any 1 6 j 6 m we have

(q λ(0)m+k,k+j−1 − q−λ

(0)m+k,k+j−1+2j−2u−1)

(q λ(0)m+k−1,k+j−1 − q−λ

(0)m+k−1,k+j−1+2j−2u−1)

∙ (q μj − q−μj+2j−2u−1)

= (q ν(j)k − q−ν

(j)k +2j−2u−1),

completing the proof of the formula for νk(u).

Similarly, using Corollary 6.10 again, we find that

ν1(u) ν2(q−2u) . . . νk(q

−2k+2u)

=(q−λ

(0)m+k,1 − q λ

(0)m+k,1 u) . . . (q−λ

(0)m+k,m+k − q λ

(0)m+k,m+k−2m−2k+2u)

(q−μ1 − q μ1u) . . . (q−μm − q μm−2m+2u),

and then proceed in the same way as in the calculation of νk(u). �

In the case where the components of λ and μ are nonnegative we may regard them as

partitions. Consider the corresponding skew diagram λ/μ and denote by c(α) the content

of a box α ∈ λ/μ so that c(α) = j − i if α occurs in row i and column j.


Corollary 9.4. The Drinfeld polynomials P1(u), . . . , Pn−1(u) corresponding to the

Uq(gln)-module L(λ)+μ are given by

Pk(u) =m+1∏

i=1

(1− q2ν(i)k+1−2i+2u)(1− q2ν

(i)k+1−2i+4u) . . . (1− q2ν

(i)k −2iu),

for k = 1, . . . , n− 1. If λ and μ are partitions then the formula can also be written as

Pk(u) =∏

α

(1− q2 c(α)u),

where α runs over the top boxes of the columns of height k in the diagram of λ/μ.

Proof. The formulae follow from (9.1) and Theorem 9.3. �

Note that if m = 0 then L(λ)+μ may be regarded as the evaluation module L(λ) over

Uq(gln). Clearly, the Drinfeld polynomials provided by Corollary 9.4 coincide with those

given by (9.7).

9.1. Gelfand–Tsetlin characters

Finally, we calculate the Gelfand–Tsetlin characters of the skew representations. Fol-

lowing [BK2, Section 5.2], introduce the set Pn of all power series of the form a(u) =

a1(u1) . . . an(un), where the u1, . . . , un are indeterminates and each ai(u) is a power series

in u−1. We shall denote by a(r)i the coefficient of u

−r of this series. Consider the group

algebra Z [Pn] of the abelian group Pn whose elements are finite linear combinations of theform

∑ma(u)[a(u)], where ma(u) ∈ Z .

We shall be working with the q-Yangian Yq(gln). Let us introduce the series hi(u) with

coefficients in Yq(gln) by the formulae

hi(u) = t1... i−11... i−1(q

2i−2u)−1 t 1... i1... i(q2i−2u), i = 1, . . . , n.

Due to (5.30), the coefficients h(r)i of all the series form a commutative subalgebra of Yq(gln)

(in fact it is a maximal commutative subalgebra), the Gelfand–Tsetlin subalgebra.

Definition 9.5. If V is a finite-dimensional representation of Yq(gln) and a(u) ∈ Pn,the corresponding Gelfand–Tsetlin subspace Va(u) consists of the vectors v ∈ V with the

property that for each i = 1, . . . , n and r > 0 there exists p > 1 such that (h(r)i −a(r)i )

p v = 0.

Then the Gelfand–Tsetlin character of V is defined by

chV =∑

a(u)∈Pn

(dimVa(u))[a(u)];

cf. [FR], [Kn].


By analogy with [BK2, Section 6.2], introduce the following special elements of the

group algebra Z [Pn] by

xi,a =[ q a+i − q−a−i u−1iq a+i−1 − q−a−i+1 u−1i

], 1 6 i 6 n, a ∈ C .

We make L(λ)+μ into a module over the q-Yangian Yq(gln) by taking the composition

of the homomorphism Yq(gln) → Yq(glm+n) given by the first relation in (6.12) with theevaluation homomorphism Yq(glm+n)→ Uq(glm+n) given by

T (u) 7→T − T u−1

1− u−1.

This additional factor, as compared to (1.45), will ensure the character formulae look

simpler. For the same purpose, we shall also assume that the components of λ and μ are

non-negative so that we can consider the corresponding skew diagram λ/μ. The formulae

in the general case will then be obtained by an obvious modification. A semistandard

λ/μ-tableau T is obtained by writing the elements of the set {1, . . . , n} into the boxes ofthe diagram of λ/μ in such a way that the elements in each row weakly increase while the

elements in each column strictly increase. By T (α) we denote the entry of T in the boxα ∈ λ/μ.

Corollary 9.6. The Gelfand–Tsetlin character of the Yq(gln)-module L(λ)+μ is given

by

chL(λ)+μ =∑

T

∏

α∈λ/μ

xT (α),c(α),

summed over all semistandard λ/μ-tableaux T .

Proof. First, we consider the particular case m = 0 so that L(λ)+μ coincides with the

evaluation module L(λ). The coefficients of the quantum determinant of Yq(gln) act on

L(λ) as scalar operators found from

qdetT (u)|L(λ) =n∏

i=1

qλi−i+1 − q−λi+i−1u−1

q−i+1 − q i−1u−1.

Observe that, regarding λ as a diagram, we can represent the product here as

∏

α∈λ

q c(α)+1 − q−c(α)−1u−1

q c(α) − q−c(α)u−1.

Note also that the quantum determinant can be factorized as

qdetT (u) = h1(u)h2(q−2u) . . . hn(q

−2n+2u).


Now we employ the well-known bijection between the patterns associated with λ and the

semistandard λ-tableaux. Namely, the pattern Λ can be viewed as the sequence of diagrams

Λ1 ⊆ Λ2 ⊆ ∙ ∙ ∙ ⊆ Λn = λ,

where the Λk are the rows of Λ. The betweenness conditions (9.2) mean that the skew

diagram Λk/Λk−1 is a horizontal strip; see, e.g., Macdonald [M, Chapter 1]. The corre-

sponding semistandard tableau is obtained by placing the entry k into each box of Λk/Λk−1.

By the above observation, for any basis vector ξΛ ∈ L(λ) and any 1 6 k 6 n we have

h1(u)h2(q−2u) . . . hk(q

−2k+2u) ξΛ =∏

α∈Λk

q c(α)+1 − q−c(α)−1u−1

q c(α) − q−c(α)u−1ξΛ,

so that

hk(q−2k+2u) ξΛ =

∏

α∈Λk/Λk−1

q c(α)+1 − q−c(α)−1u−1

q c(α) − q−c(α)u−1ξΛ. (9.15)

Thus, the element of Z [Pn] corresponding to the eigenvalue of hk(uk) on ξΛ coincides withthe product ∏

α∈Λk/Λk−1

xk,c(α).

This proves the claim for the case m = 0.

In the skew case (m > 1) we use Corollary 6.10 which implies that formula (9.15)remains valid when the action is considered on the basis vector ζΛ ∈ L(λ)

+μ instead of ξΛ.

Here we denote by Λk the row (λm+k,1, . . . , λm+k,m+k) of the trapezium pattern Λ and use a

natural bijection between the trapezium patterns and the semistandard λ/μ-tableaux. �

Remark 9.7. Note that the Gelfand–Tsetlin character of the corresponding skew mod-

ule L(λ)+μ over the Yangian Y(gln) is given by the same formula as in Corollary 9.6, where

xi,a is now defined by xi,a = [1 + (ui + a + i − 1)−1]; see [BK2, Section 6.2]. The skewrepresentations of the Yangians and quantum affine algebras were studied in the litera-

ture from various viewpoints providing different interpretations of the character formula

of Corollary 9.6; see e.g. [BR], [C], [Mo], [NN], [NT2]. For the evaluation modules, a

calculation similar to the above can be found e.g. in [FM, Section 4.5], [BK2, Section 7.4].

One can easily extend the definition of the Gelfand–Tsetlin character to representations

of Uq(gln) by considering the commutative subalgebra generated by the coefficients of the

series hi(u) together with the coefficients of the hi(u) which are defined in the same way with

the use of the quantum minors of T (u). However, by the results of [FR], the eigenvalues of

the hi(u) on finite-dimensional representations are essentially determined by those of hi(u).

In particular, the character formula for the Uq(gln)-module L(λ)+μ will have the form given

in Corollary 9.6. �

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